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THEORETICAL INVESTIGATION OF THE EFFECT OF MEMBRANE PROPERTIES

OF NANOPOROUS RESERVOIRS ON THE PHASE BEHAVIOR

OF CONFINED LIGHT OIL

by

Ziming Zhu

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ii

A thesis submitted to the Faculty and the Board of Trustees of the Colorado School of

Mines in partial fulfillment of the requirements for the degree of Master of Science

(Petroleum Engineering).

Golden, Colorado

Date _____________________________

Signed: _____________________________ Ziming Zhu

Signed: _____________________________

Dr. Xiaolong Yin Thesis Advisor

Signed: _____________________________

Dr. Erdal Ozkan Thesis Advisor

Golden, Colorado

Date _____________________________

Signed: _____________________________ Dr. Erdal Ozkan

Professor and Department Head Department of Petroleum Engineering

iii

ABSTRACT

This theoretical study is an effort to probe the effect of semi-permeable membrane

properties of a nanoporous medium on hydrocarbon phase behavior in tight-oil reservoirs.

It is assumed that the fluids stored in a nanoporous reservoir are divided into two parts:

one part that is already filtered and can flow to a production well without compositional

change, and another part that replenishes the filtered fluid according to the filtration

efficiency of the nanoporous medium and the prevailing filtration pressure. This selective

hydrocarbon transport leads to a pressure difference between the unfiltered and filtered

parts of the porous medium as well as significant compositional changes in the filtered

and unfiltered parts. The compositional change, fluid density, viscosity and interfacial

tension are calculated as functions of pressure when the depletion pressure decreases

below the bubble point pressure of the filtered part. Through simulating a pressure

depletion of a porous medium with internal filtration, we find that membrane filtration

makes the produced hydrocarbon mixture lighter, and traps the heavier components in

the reservoir. These findings and results can help us better understand and characterize

the behavior of reservoir fluids during pressure depletion, and may provide us new

perspectives for potential EOR applications.

iv

TABLE OF CONTENTS

ABSTRACT ...………………………………………………………………………………......iii

LIST OF FIGURES ..........................................................................................................vi

LIST OF TABLES .......................................................................................................... viii

ACKNOWLEDGEMENT ..................................................................................................xi

CHAPTER 1 INTRODUCTION ..................................................................................... 1

1.1 Problem Statement........................................................................................... 1

1.2 Objectives ........................................................................................................ 1

1.3 Current Solutions ............................................................................................. 2

1.4 Thesis Organization ......................................................................................... 3

CHAPTER 2 BACKGROUND ...................................................................................... 4

2.1 Chemical Osmosis ........................................................................................... 4

2.1.1 Osmotic Equilibrium and Osmotic Pressure .......................................... 4

2.1.2 Osmotic Efficiency ...................................................................................... 6

CHAPTER 3 RESERVOIR FLUIDS PHASE EQUILIBRIUM CALCULATION ............ 11

3.1 Peng-Robinson Equation of State .................................................................. 11

3.2 Single-Phase Equilibrium Property Calculation ............................................... 12

3.3 Vapor-Liquid Two-Phase Equilibrium Calculation and Interfacial Tension ...... 16

3.4 Liquid Viscosity Calculation by Lohrenz Correlation ....................................... 18

CHAPTER 4 SIMULATION OF MEMBRANE FILTRATION IN HYDROCARBON SATURATED NANOPOROUS MEDIA ................................................. 21

4.1 Hydrocarbon Filtration Model Development .................................................... 21

4.2 Filtration Equilibrium and Filtration Pressure .................................................. 22

4.3 Fugacity-Based Filtration Efficiency Calculation ............................................. 24

v

4.4 Single-Solute and Multi-Solute Filtration Efficiency Calculation ...................... 25

CHAPTER 5 SIMULATION OF PRESSURE DEPLETION OF A SINGLE-CELL

RESERVOIR WITH AN INTERNAL MEMBRANE ................................. 29

5.1 Establishing Initial-Equilibrium before Pressure Depletion .............................. 30

5.2 Computing Post-Initial Equilibriums during Pressure Depletion ...................... 33

CHAPTER 6 RESULTS AND DISCUSSIONS ........................................................... 37

CHAPTER 7 SUMMARY, CONCLUSIONS AND RECOMMENDATIONS FOR

FUTURE WORK ................................................................................... 66

LIST OF SYMBOLS ...................................................................................................... 69

REFERENCES CITED .................................................................................................. 74

APPENDIX A SIMULATION OF A PRESSURE DEPLETION PROCESS USING

THE FUGACITY-BASED FILTRATION EFFICIENCY .......................... 77

A.1 Solution of the Osmotic Pressure – Membrane Efficiency Equation ............... 77

A.2 Coupling Membrane Filtration with Pressure Depletion .................................. 80

A.3 Simulation Results ......................................................................................... 82

APPENDIX B SIMULATION CASE WITH A FILTRATION EFFICIENCY OF

[0.75, 0.9] .............................................................................................. 93

vi

LIST OF FIGURES

Figure 2.1 Osmotic equilibrium and osmotic pressure illustration: (a) Initial condition (b) Equilibrium condition. (Geren et al. 2014) ................................................ 5

Figure 2.2 (a) Electrical charge distribution near a clay surface. (b) Electrical potential profile in a wide gap. (c) Electrical potential profile in a narrow gap. (Mitchell 2005; Keijzer 2000) ....................................................................................... 9

Figure 2.3 Hydrocarbon molecular diameter. (Nelson 2009) ........................................... 9

Figure 3.1 Single-phase equilibrium calculation flow chart. ........................................... 13

Figure 3.2 Vapor-liquid two-phase equilibrium calculation flow chart. ........................... 17

Figure 4.1 Dual-pore filtration model. ............................................................................ 21

Figure 4.2 Equilibrium state of dual-pore filtration model. ............................................. 23

Figure 5.1 Two-part single-cell reservoir model: Initial-equilibrium before pressure depletion. .................................................................................................... 30

Figure 5.2 Computational procedure of initial-equilibrium stage. ................................... 31

Figure 5.3 Two-part single cell reservoir model: Post-initial equilibrium during pressure depletion. ...................................................................................................... 33

Figure 5.4 Computational procedure of post-initial equilibrium stage. ........................... 34

Figure 6.1 Pressure difference between filtered and unfiltered part at different depletion pressures for different cases. ....................................................................... 47

Figure 6.2 Molecule weight of fluid in unfiltered/filtered part at different depletion pressures, ideal membrane. ........................................................................ 49

Figure 6.3 Molecule weight of fluid in unfiltered/filtered part at different depletion pressures, non-ideal membrane. ................................................................ 50

Figure 6.4 Vapor phase molar fraction in filtered part at different depletion pressures for different cases. ....................................................................................... 54

vii

Figure 6.5 Vapor phase molar fraction in filtered part at different depletion pressures for different cases, validation results from winprop. ..................................... 55

Figure 6.6 Viscosity of liquid in unfiltered/filtered part at different depletion pressures for different cases. ....................................................................................... 58

Figure 6.7 Viscosity of liquid in unfiltered/filtered part at different depletion pressures for different cases, validation results from winprop. ..................................... 59

Figure 6.8 Vapor/Liquid interfacial tensions in filtered part at different depletion pressures for different cases. ..................................................................... 61

Figure 6.9 Interfacial tension vs. Pressure for various reservoir oils (Firoozabadi et al. 1988). ........................................................................................................... 62

Figure 6.10 Density of liquid phase in filtered part at different depletion pressures for different cases. ........................................................................................... 64

Figure 6.11 Density of vapor phase in filtered part at different depletion pressures for different cases. ........................................................................................... 65

Figure A.1 Dual-pore system used to calculate membrane efficiency. (geren 2014) .... 77

Figure A.2 Computational procedure of solving the osmotic pressure-membrane efficiency equation. (geren et al. 2014) ...................................................... 79

Figure A.3 Three-pore system used to simulate the coupling of membrane filtration with pressure depletion. ............................................................................. 80

Figure A.4 Computational procedure of coupling membrane filtration with pressure depletion. .................................................................................................... 82

Figure A.5 Osmotic pressure at different depletion pressures. ...................................... 92

viii

LIST OF TABLES

Table 5.1 Known (√) and unknown (×) variables for the initial-equilibrium stage........... 32

Table 5.2 Known (√) and unknown (×) variables for the post-initial equilibrium stage ... 35

Table 6.1 Thermodynamic parameters of components in the light oil ........................... 37

Table 6.2 Initial state parameters .................................................................................. 37

Table 6.3 The initial-equilibrium state before pressure depletion .................................. 38

Table 6.4 Post-initial equilibrium state when the pressure of the filtered part is reduced to 30 psi. Ideal membrane σ = [1, 1] ............................................................. 40

Table 6.5 Post-initial equilibrium state when the pressure of the filtered part is reduced to 30 psi. Non-ideal membrane σ = [0.35, 0.55]............................................ 41

Table 6.6 Post-initial equilibrium state when the pressure of the filtered part is reduced to 30 psi. Non-selective membrane, σ = [0, 0] .............................................. 42

Table 6.7 Overall and individual filtration efficiencies for every case ............................ 43

Table 6.8 Pressure of unfiltered/filtered part at different depletion pressures, ideal membrane case ............................................................................................ 46

Table 6.9 Pressure of unfiltered/filtered part at different depletion pressures, non-ideal membrane case ............................................................................................ 46

Table 6.10 Molecule weight of fluid in unfiltered/filtered part at different depletion pressures, ideal membrane case ............................................................... 48

Table 6.11 Molecule weight of fluid in unfiltered/filtered part at different depletion pressures, non-ideal membrane case ........................................................ 48

Table 6.12 Vapor phase molar fraction in filtered part at different depletion pressures for different cases ........................................................................................ 53

Table 6.13 Vapor phase molar fraction in filtered part at different depletion pressures for different cases, validation results from WinProp .................................... 53

ix

Table 6.14 Viscosity of liquid in unfiltered/filtered part at different depletion pressures for different cases ....................................................................................... 56

Table 6.15 Viscosity of liquid in unfiltered/filtered part at different depletion pressures for different cases, validation results from WinProp .................................... 57

Table 6.16 Vapor/Liquid interfacial tensions in filtered part at different depletion pressures for different cases ..................................................................... 60

Table 6.17 Density of fluids in filtered part at different depletion pressures for different cases ........................................................................................................... 63

Table A.1 Thermodynamic model parameters of the components in the light oil .......... 83

Table A.2 Simulation parameters and results before pressure depletion ...................... 84

Table A.3 Fluid properties when the pressure of System II is reduced to 45 psi. Membrane effect is implemented between System I and System II ............ 85

Table A.4 Fluid properties from a constant composition expansion at 45 psi without membrane filtration ....................................................................................... 87

Table A.5 Fluid Properties when the pressure of System II is reduced to 35 psi. Membrane effect is included ....................................................................... 88

Table A.6 Fluid properties when the pressure of System II is reduced to 35 psi. Membrane effect is not included ................................................................. 89

Table A.7 Fluid properties when the pressure of System II is reduced to 25 psi. Membrane effect is included ....................................................................... 90

Table A.8 Fluid properties when the pressure of System II is reduced to 25 psi. Membrane effect is not included ................................................................. 91

Table B.1 Thermodynamic parameters of components in the light oil ........................... 93

Table B.2 Initial state parameters .................................................................................. 93

Table B.3 The initial-equilibrium state before pressure depletion .................................. 94

Table B.4 Post-initial equilibrium state when the pressure of the filtered part is reduced to 30 psi. Ideal membrane, σ = [1, 1] ............................................................ 95

x

Table B.5 Post-initial equilibrium state when the pressure of the filtered part is reduced to 30 psi. Non-ideal membrane, σ = [0.75, 0.9] ............................................ 96

Table B.6 Post-initial equilibrium state when the pressure of the filtered part is reduced to 30 psi. Non-selective membrane, σ = [0, 0] .............................................. 97

Table B.7 Overall and individual filtration efficiencies for every case ............................ 98

xi

ACKNOWLEDGEMENT

I would like to thank my advisors Dr. Xiaolong Yin and Dr. Erdal Ozkan, who have

provided the best guidance and mentorship for me. Their wisdom, vision, rigorous attitude

and exploration spirit have influenced me profoundly, and I learned a lot from them. Also,

I am grateful that they have broadened my horizon in the industry by creating many

valuable opportunities for me.

I would like to thank my committee members Dr. Hazim H. Abass and Dr. Manika

Prasad for their contributions and guidance. Also, I am thankful to Dr. Lei Wang for his

support and encouragement.

I would like to thank all my friends and colleagues at Colorado School of Mines,

because of you all, I had a memorable and fulfilling experience at Golden.

Finally, I want to thank my parents and my brother for their unconditional support

and endless love.

1

CHAPTER 1

INTRODUCTION

This chapter introduces the problem of hydrocarbon filtration, the objectives of this

thesis and current solutions, and the organization of this thesis.

1.1 Problem Statement

As hydrocarbons in tight oil reservoirs are primarily stored in nano-sized pores, a

particular issue when the sizes of pores and pore throats decrease down to the sizes of

hydrocarbon molecules is that nanoporous reservoirs may display membrane properties,

acting like a semi-permeable membrane that permits certain molecules to pass through

freely yet restricts the transport of other components with larger diameters. This

membrane property of nanoporous reservoir will result in compositional differences

between different parts of the reservoir and unbalanced pressures even at the equilibrium

condition. Therefore, when the membrane effect is present, the phase behavior of

reservoir fluids may deviate from the ordinary case, and it is not adequate to use a single

composition to characterize the reservoir fluids for the entire nanoporous reservoir.

1.2 Objectives

As stated, the presence of nanoporous reservoir membrane properties may

generate a compositional difference between different parts of the reservoir even at the

equilibrium condition. As a result, it is not appropriate to characterize the reservoir fluids

properties and behavior by using a single composition. To better understand the effect of

nanoporous reservoir membrane properties on reservoir fluids and the behavior of a

nanoporous reservoir during depletion, we set the objectives of this work as follows.

2

Establish filtration mechanisms and introduce filtration model equations that can

quantitatively describe the effect of membrane properties on multicomponent

phase equilibrium based on the chemical osmosis theory.

Develop a numerical simulator to solve the above equations and simulate the

phase behavior, fluid properties and transfer of reservoir fluids in a single-cell

nanoporous petroleum system with membrane properties during a pressure

depletion.

1.3 Current Solutions

The study of Geren et al. (2014) is the first work, where the equilibrium

compositions and pressures of hydrocarbon mixtures were computed across a semi-

permeable membrane, using a model that defines the filtration efficiency based on the

fugacity difference of the filtrated component. It was found, as anticipated, that the

fugacity difference across the membrane for the filtrated component generated a pressure

difference across the membrane at the equilibrium state. In this study, we extended the

study of Geren et al. (2014) on membrane equilibrium to a theoretical modeling of the

phase behavior of a light oil, confined in a porous medium with a semi-permeable

membrane, during a constant composition expansion process. The porous medium,

together with the fluid therein, is divided into two parts: one part that is already “filtered”

and can flow to a production well without compositional change, and another part that

replenishes the “filtered” fluids through a semi-permeable membrane. Except for the

pressure and compositional differences across the membrane, there is no other pressure

and compositional variations within the porous medium. The system, therefore, is

effectively a single-cell reservoir under pressure depletion. The depletion process was

3

first modeled using the filtration efficiency model of Geren et al. (2014). The work was

published in Zhu et al. (2015). Since then, another filtration efficiency model analogous

to the theory of chemical osmosis was developed. The thesis primarily covers the new

filtration efficiency model and its predictions. The equilibrium compositions of

multicomponent hydrocarbon mixtures separated by the semi-permeable membrane

were calculated using a hypothetical filtration efficiency σ (0 < σ < 1) and the modified

Peng-Robinson equation of state during the pressure depletion process. Additionally, the

changes in fluid properties such as density, viscosity, interfacial tension were calculated

and reported. By comparing these properties to those computed without the membrane,

the effect of membrane properties on reservoir fluid properties is shown.

1.4 Thesis Organization

This thesis is organized as follows: Chapter 1 introduces the problem, the

objectives, and the scope of the thesis. Chapter 2 describes the theoretical backgrounds,

including chemical osmosis theory and a discussion on filtration mechanisms of shale.

Chapter 3 elaborates on the reservoir fluid phase equilibrium calculation procedures,

including single-phase equilibrium property calculations and vapor-liquid two-phase

equilibrium calculations. Chapter 4 describes our approach to model membrane filtration

in a hydrocarbon-saturated dual-pore medium, by introducing the filtration model, the

concepts of filtration equilibrium and filtration pressure, and the single-solute and multi-

solute filtration efficiency calculation steps. Chapter 5 presents a simulation of pressure

depletion of a single-cell nanoporous reservoir with membrane properties and describes

the related calculation procedures. Chapter 6 gives the simulation results and discussions.

Chapter 7 presents the summary, conclusions, and recommendations for future work.

4

CHAPTER 2

BACKGROUND

This chapter presents the chemical osmosis theory, including osmotic equilibrium,

osmotic pressure, and osmotic efficiency. After that, the filtration mechanisms of shale

are discussed.

2.1 Chemical Osmosis

Chemical osmosis is the spontaneous passage or diffusion of solvent molecules

through a semi-permeable membrane, which selectively allows the passage of solute

molecules, into the region with greater solute concentration, in the direction that tends to

equalize the solute concentrations of the two sides. This chemical process was introduced

in 1854 by a British chemist, Thomas Graham, and first thoroughly studied in 1877 by a

German plant physiologist, Wilhelm Pfeffer. (Cath et al. 2006)

2.1.1 Osmotic Equilibrium and Osmotic Pressure

As shown in Figure 2.1 (a), a low-salinity solution and a high-salinity solution in the

U-shape tube are initially separated by an idealized membrane in the middle of the tube.

Water molecules on both sides can flow freely in both directions through the idealized

membrane freely. Solute molecules are completely restricted from passing through the

membrane. Due to the solute concentration difference between the two sides, solvent

molecules will move from the low-concentration side to the high-concentration side in an

attempt to equalize the solute concentration. This flow of solvent constitutes an osmotic

flow. As the solvent molecules continue flowing from the low-concentration side to the

high-concentration side, the hydrostatic pressure on the high-concentration side

increases. The pressure difference will generate a tendency for the solvent molecules to

5

move from the high-concentration side to the low-concentration side, in the direction

opposite to the osmotic flow. Given sufficient equilibration time, the pressure-driven flow

of the solvent will eventually balance the concentration-driven (chemical potential-driven)

osmotic flow, and the system reaches osmotic equilibrium. The pressure difference

needed to establish osmotic equilibrium is defined as the osmotic pressure.

Figure 2.1 Osmotic equilibrium and osmotic pressure illustration: (a) Initial condition (b) Equilibrium condition. (Geren et al. 2014)

The Dutch physical and organic chemist, also the first winner of the Nobel Prize in

Chemistry, Jacobus H. van ’t Hoff, discovered the relationship between osmotic pressure

and temperature and its analogy to the ideal gas law (van ‘t Hoff 1995). In 1886, van ’t

Hoff derived the osmotic equation, which is applicable to calculate the osmotic pressure

π of an ideal solution with low solute concentration. Below is the revised van ’t Hoff

osmotic equation.

π = 𝑖𝑀𝑅𝑇 (2.1)

6

Where i is the dimensionless van ‘t Hoff factor, which describes the dissociation and

association of the solute in the solution. M is the molarity of the solutes. R is the gas

constant, and T is the temperature. This equation gives the absolute osmotic pressure of

a solution when it is separated from the pure solvent by a semi-permeable membrane.

When a semi-permeable membrane lies in the middle of two solutions with different

concentrations, the total osmotic pressure on the membrane is given by the difference

between the osmotic pressures on the two sides.

2.1.2 Osmotic Efficiency

Through an ideal membrane, the fluid flux only consists of solvent, whereas

through a non-ideal membrane the fluid flux also includes some solute components.

Therefore, for the same original solution, the observed osmotic pressure for a system with

a non-ideal membrane will be different from that with an ideal membrane. The non-ideality

of a membrane is described by its osmotic efficiency, which is defined as the observed or

realistic osmotic pressure 𝜋𝑟𝑒𝑎𝑙, divided by the theoretical osmotic pressure 𝜋𝑖𝑑𝑒𝑎𝑙.

Staverman (1952) initially termed this ratio as the “reflection coefficient”, σ, which

is expressed by the equation below,

𝜎 = ( 𝜋𝑟𝑒𝑎𝑙

𝜋𝑖𝑑𝑒𝑎𝑙)𝐽𝑣=0 (2.2)

Where 𝜋𝑟𝑒𝑎𝑙 is the observed osmotic pressure, 𝜋𝑖𝑑𝑒𝑎𝑙 is the theoretical osmotic pressure

across the membrane and 𝐽𝑣 is the net fluid flux through the membrane. The osmotic

efficiency 𝜎 is measured at the equilibrium state, which is when the net fluid flux is zero.

As this osmotic efficiency can be used to quantitatively characterize the ability of a

material acting as a semi-permeable membrane, it is also referred as the membrane

efficiency or filtration efficiency. The value of filtration efficiency ranges between zero, for

7

a non-selective membrane, and one, for an ideal membrane. Membranes with values in

between are called non-ideal membranes.

2.2 Filtration Mechanisms of Shale

Shale is a fine-grained, clastic sedimentary rock composed of mud that is a mix of

flakes of clay minerals and tiny fragments (silt-sized particles) of other minerals,

especially quartz and calcite.

Studies of the osmotic flow of water through samples of shale and siltstone have

indicated that shale filters salt from solution (McKelvey and Milne 1960; Young and Low

1965). Later, Magara (1974) showed the inverse relationship between the salinity

distribution and pore sizes in shale due to the ion-filtration effect of shale. More recently,

Neuzil (2000) confirmed the significant role of membrane properties of shale through a

nine-year in-situ measurement of the pressure of the fluid and solute concentration in the

Cretaceous-age Pierre Shale in South Dakota. Garavito et al. (2006) numerically modeled

the fluid pressures and concentrations obtained in Neuzil’s experiment and verified the

generation of large (up to 20 MPa) osmotic pressure anomalies. Other researchers (Revil

and Pessel 2002) discussed the electro-osmotic flow of pore water in nanopores due to

electrical potential gradient created by various natural phenomena.

The evidence of membrane properties of shale for hydrocarbons is derived from

the observed compositional differences between hydrocarbons in the reservoir and its

associated source rocks. Brenneman and Smith (1958), Hunt and Jameson (1956), and

Hunt (1961) all noted that most of the source oils are composed of more aromatic

hydrocarbons when they are compared with their reservoir oils. These observations point

to some level of sieving for hydrocarbon molecules in nanoporous shales.

8

The membrane properties of shale manifest themselves in two forms: Electrostatic

Exclusion and Steric Hindrance. Below, we briefly describe these two filtration

mechanisms.

Electrostatic Exclusion

Clay contents of shale are often larger than 50% (Prasad 2012), and clay minerals

are usually negatively charged. The electrical double layer (EDL) of adjacent clay

platelets could explain the membrane property of shale on charged solutes in aqueous

solutions. The effective thickness of EDL, which is the Debye-Hückel length 𝑘−1, can

range between tens of Å and few of micrometers (Weaver 1989), depending on the ionic

concentration of the solution. In fresh waters where the cation concentration is low, the

double layer commonly has a thickness in excess of 30 to 70 Å (Weaver 1989) and the

approximate thickness calculated for electrolyte concentrations of 0.001 M is 10 nm. In

Figure 2.2 (a), the negatively charged clay surface repels anions and attracts cations to

stay near the clay surface to maintain electrical neutrality, forming an EDL. As a result,

ions attempting to pass by clay platelets will be restricted. Usually, in conventional

reservoirs, as the thickness of EDL is insignificant in relation to the pore size, the EDLs

of clay platelets do not overlap with each other; as shown in Figure 2.2 (b), there is a

neutral zone, within which both charged species and uncharged species could move in

and out freely. Conversely, in unconventional reservoirs, EDLs may overlap each other

to varying degrees within the nanopore space, as shown in Figure 2.2 (c). In this case,

charged species will be prevented from passing through, and only the uncharged species

can pass through the pore throat. (Mitchell 2005; Keijzer 2000; Marine and Fritz 1981).

9

Figure 2.2 (a) Electrical charge distribution near a clay surface. (b) Electrical potential profile in a wide gap. (c) Electrical potential profile in a narrow gap. (Mitchell 2005;

Keijzer 2000)

Steric Hindrance

According to the IUPAC pore size classification (Sing 1985), unconventional

reservoir pores can be divided into three broad categories:

Micropores: pores with pore-width below 2 nm.

Mesopores: pores with pore-width between 2 nm and 50 nm.

Macropores: pores with pore-width greater than 50 nm.

Figure 2.3 Hydrocarbon molecular diameter. (Nelson 2009)

10

Based on the pore size distribution in shale, a fraction of pores can be classified as

micropores, which have a diameter less than 2 nm and an even smaller pore throat size.

The volume fraction of micropores in some shale samples ranges around 9%, and can

be as high as 19.23% (Kuila and Prasad 2011). On the other hand, the molecular sizes

of paraffin, aromatics and asphaltenes lie between 0.5 nm and 10 nm (Nelson 2009). It is

therefore expected that some steric hindrance should occur when the molecular size of

hydrocarbon components becomes comparable to or even exceeds the pore throat size.

In this situation, the solvents, the components with lower molecular weights, can pass

through the restriction, whereas the solute, the components with higher molecular weights,

will be restricted at the pore throats or even forbidden to pass through.

11

CHAPTER 3

RESERVOIR FLUIDS PHASE EQUILIBRIUM CALCULATION

This chapter presents procedures of single-phase equilibrium property calculation,

vapor-liquid two-phase equilibrium calculation, interfacial tension and liquid viscosity

calculation. All of the equilibrium calculations applied in this thesis are based on Peng-

Robinson equation of state.

3.1 Peng-Robinson Equation of State

Reservoir fluids simulations usually employ equilibrium calculations to calculate the

number of equilibrium phases, the compositions, and the mole fraction of each phase.

Cubic equations of state (EOS) are widely used for the calculation of phase equilibrium

because of their accuracy, simplicity, and solvability. Peng-Robinson equation of state

(PR-EOS) is one of the most commonly used cubic EOS in the petroleum industry. PR-

EOS can provide reasonable accuracy near the critical point, particularly for calculations

of the compressibility factor and liquid phase density. Additionally, this equation applies

to all calculations of all fluid properties in natural gas processes (Peng and Robinson

1976).

In this study, PR-EOS is applied to single-phase and two-phase equilibrium

property calculations. Admittedly, application of PR-EOS in a confined environment is

probably not justified; it is not the most accurate EOS for two-phase equilibrium

calculations, either. However, it should be sufficient for our preliminary model, the intent

of which is to illustrate the process of pressure depletion with membrane filtration

qualitatively in the absence of experimental data.

12

3.2 Single-Phase Equilibrium Property Calculation

A Peng-Robinson equation of state based single-phase equilibrium property

calculation is used to compute the properties of reservoir fluids in this thesis. The

calculation procedure is elaborated in the following paragraphs. The calculation flow chart

is shown in Figure 3.1.

For hydrocarbon mixtures stored in a reservoir of temperature T and pressure P,

judging whether they are at a single- or two-phase state requires trial equilibrium

calculations that need the equilibrium ratios 𝐾𝑖 =𝑦𝑖

𝑥𝑖, the ratio between the mole fractions

of component i in the vapor and liquid phases. To start this calculation, initial values of 𝐾𝑖

from the Wilson equation (Wilson 1968) are used.

𝐾𝑖 =1

P𝑟,𝑖exp [5.37(1 + ωi)(1 −

1

T𝑟,𝑖)] (3.1)

ωi is the acentric factor, 𝑃𝑟,𝑖 and 𝑇𝑟,𝑖 are the reduced pressure and temperature of

component i. Then, the Rachford-Rice equation is solved to obtain the fractions of the

vapor and liquid phases.

𝑓(1 − 𝑁𝑜) = ∑ [(𝐾𝑖−1)𝑍𝑖

(𝐾𝑖−1)(1−𝑁𝑜)+1]𝑛𝑐

𝑖=1 (3.2)

𝑁𝑜 is the mole fraction of the liquid phase in the whole mixture, 𝑍𝑖 is the overall

composition of component i and nc is the number of components of the fluids. According

to the value of 𝑓(1 − 𝑁𝑜), the phase status can be determined as below:

If 𝑓(1) < 0, mixture is all liquid.

If 𝑓(0) > 0, mixture is all vapor.

If 𝑓(1) > 0 and 𝑓(0) < 0, mixture is in two-phase status (liquid + vapor).

13

Then, the phase composition respective to each of the three phase states above is

calculated. Here we use a liquid-phase case to illustrate the calculation procedure.

Figure 3.1 Single-phase equilibrium calculation flow chart.

END

Y

𝑦𝑖 = 1 𝑥𝑖 = 1

Calculate z, 𝜙, 𝑓

𝑅𝑖 =𝑓𝑖

𝐿

𝑓𝑖𝑉

ȁ𝑅𝑖 − 1ȁ ≤ 10−6 Update

𝐾𝑖′ = 𝑅𝑖𝐾𝑖

N

Y Y

𝑁𝑜 = 1

𝑥𝑖 = 𝑍𝑖

𝑦𝑖 = 𝐾𝑖𝑍𝑖

𝑁𝑜 = 0

𝑦𝑖 = 𝑍𝑖

𝑥𝑖 = 𝑍𝑖/𝐾𝑖

f(1)<0 f(0)>0

𝑓(1 − 𝑁𝑜) = ቈ(𝐾𝑖 − 1)𝑍𝑖

(𝐾𝑖 − 1)(1 − 𝑁𝑜) + 1

𝑛𝑐

𝑖=1

START

𝐾𝑖 =1

P𝑟,𝑖exp ቈ5.37(1 + ωi)(1 −

1

T𝑟,𝑖)

14

If mixture is all liquid phase, that is 𝑓(1) < 0,

𝑁𝑜 = 1, 𝑥𝑖 = 𝑍𝑖 , 𝑦𝑖 = 𝐾𝑖 ∗ 𝑍𝑖

Note that: here we still calculate the composition of the “vapor phase” that should not exist

in an all-liquid mixture. The reason is that the composition of the “vapor phase” is

necessary for the iteration.

After obtaining the composition of the liquid and vapor phases, the next step is to

calculate the compressibility factor z, the fugacity coefficient 𝜑 𝑖 and fugacity 𝑓𝑖 for every

component in the two phases. Here, the cubic Peng-Robinson EOS is applied to calculate

the compressibility factor.

𝑧3 − (1 − 𝐵)𝑧2 + (𝐴 − 3𝐵2 − 2𝐵)𝑧 − (𝐴𝐵 − 𝐵2 − 𝐵3) = 0 (3.3)

where

𝐴 =𝑎𝑃

𝑅2𝑇2 = [∑ ∑ 𝑎𝑖𝑗𝑥𝑖𝑛𝑐𝑛=1

𝑛𝑐𝑖=1 𝑥𝑗]

𝑃

𝑅2𝑇2 (3.4)

𝐵 =𝑏𝑃

𝑅𝑇= [∑ 𝑏𝑖𝑥𝑖

𝑛𝑐𝑖=1 ]

𝑃

𝑅𝑇 (3.5)

Here, 𝑥𝑖 is the mole fraction of component i in the liquid phase. For the gas phase, 𝑦𝑖,

the mole fraction of component i in the gas phase is used.

𝑎𝑖𝑗 = (1 − 𝛿𝑖𝑗)𝑎𝑖

1

2𝑎𝑗

1

2 (3.6)

Here, 𝛿𝑖𝑗 is the binary interaction coefficient between component i and j.

𝑎𝑖 = [Ω𝑎𝑅2𝑇𝑐,𝑖

2

𝑃𝑐,𝑖] ቈ1 + 𝑘𝑖(1 − 𝑇

𝑟,𝑖

1

2 )

2

(3.7)

𝑏𝑖 = Ω𝑏𝑅𝑇𝑐,𝑖

𝑃𝑐,𝑖 (3.8)

15

𝑘𝑖 = 0.37464 + 1.54226𝜔𝑖 + 0.26992𝜔𝑖2 (3.9)

Here, Ω𝑎 and Ω𝑏 are the constants used in Peng-Robinson equations, and 𝜔𝑖 is the

acentric factor of component i.

Ω𝑎 = 0.457235

Ω𝑏 = 0.077796

The fugacity coefficients of component i in the liquid and vapor phases are calculated

respectively by using the following equations:

𝑙𝑛𝜙𝑖𝐿 =

𝑏𝑖

𝑏(𝑧𝐿 − 1) − ln(𝑧𝐿 − 𝐵) −

𝐴

2√2𝐵{(

2 ∑ 𝑥𝑗𝑎𝑗𝑖𝑛𝑐𝑗=1

𝑎−

𝑏𝑖

𝑏) 𝑙𝑛 (

𝑧𝐿+(√2+1)𝐵

𝑧𝐿−(√2−1)𝐵)} (3.10)

𝑙𝑛𝜙𝑖𝑉 =

𝑏𝑖

𝑏(𝑧𝑉 − 1) − ln(𝑧𝑉 − 𝐵) −

𝐴

2√2𝐵{(

2 ∑ 𝑦𝑗𝑎𝑗𝑖𝑛𝑐𝑗=1

𝑎−

𝑏𝑖

𝑏) 𝑙𝑛 (

𝑧𝑉+(√2+1)𝐵

𝑧𝑉−(√2−1)𝐵)}(3.11)

The fugacities of component i in the liquid and vapor phases are related to their respective

fugacity coefficients by:

𝑓𝑖𝐿 = 𝜙𝑖

𝐿𝑥𝑖𝑃 (3.12)

𝑓𝑖𝑉 = 𝜙𝑖

𝑉𝑦𝑖𝑃 (3.13)

For a regular vapor-liquid system, when equilibrium is achieved, the fugacity of every

component in the liquid phase and vapor phase should be identical.

𝑓𝑖𝐿 = 𝑓𝑖

𝑉 (3.14)

Here, we define the fugacity ratio 𝑅𝑖 =𝑓𝑖

𝐿

𝑓𝑖𝑉 to help evaluate Eq. (3.14). If 𝑅𝑖 satisfies the

error tolerance ȁ𝑅𝑖 − 1ȁ ≤ 10−6 for every component i, the system is treated as an

equilibrium system and the iteration is terminated. If the error tolerance is not satisfied,

the values of 𝐾𝑖 are updated by the equation below, and the calculations are iterated for

another round.

𝐾𝑖′ = 𝑅𝑖𝐾𝑖 (3.15)

16

3.3 Vapor-Liquid Two-Phase Equilibrium Calculation and Interfacial Tension

From the previous section, when the Rachford-Rice equation satisfies 𝑓(1) > 0

and 𝑓(0) < 0, the mixture should consist of vapor and liquid phases. In the two-phase

region, by using the equilibrium ratio 𝐾𝑖 , the liquid phase molar fraction 𝑁𝑜 can be

calculated from

∑ [(𝐾𝑖−1)𝑍𝑖

𝑁𝑜+𝐾𝑖(1−𝑁𝑜)]𝑛𝑐

𝑖=1 = 0 (3.16)

Afterwards, 𝑥𝑖 and 𝑦𝑖 can be obtained by

𝑥𝑖 =𝑍𝑖

1+(1−𝑁𝑜)×(𝐾𝑖−1) (3.17)

𝑦𝑖 = 𝑥𝑖 × 𝐾𝑖 (3.18)

The fugacities of component i in the liquid and vapor phases can be then calculated

respectively by Eq. (3.12) and Eq. (3.13). If the fugacities of component i are not equal

across phases, the equilibrium ratios will be updated by Eq. (3.19) and the iteration

continues. The process stops when the fugacities of component i in the two phases

become equal.

𝐾𝑖′ =

𝜙𝑖𝐿

𝜙𝑖𝑉 (3.19)

In Eq. (3.19), 𝜙𝑖𝐿 and 𝜙𝑖

𝑉 are the fugacity coefficients of component i in the liquid and

vapor phases, respectively.

After the iteration finds the vapor-liquid equilibrium, the interfacial tension (IFT)

between the vapor and liquid phases is calculated by using the Weinaug-Katz (1943)

equation.

17

Figure 3.2 Vapor-Liquid two-phase equilibrium calculation flow chart.

IFT

𝑓𝑖𝐿 𝑓𝑖

𝑉

Y

END

𝑓(1 − 𝑁𝑜) = ቈ(𝐾𝑖 − 1)𝑍𝑖

(𝐾𝑖 − 1)(1 − 𝑁𝑜) + 1

𝑛𝑐

𝑖=1

START

𝐾𝑖 =1

P𝑟,𝑖exp ቈ5.37(1 + ωi)(1 −

1

T𝑟,𝑖)

f(1)>0 & f(0)<0

Y

ቈ(𝐾𝑖 − 1)𝑍𝑖

𝑁𝑜 + 𝐾𝑖(1 − 𝑁𝑜)

𝑛𝑐

𝑖=1

= 0

𝑦𝑖 𝑥𝑖

𝑓𝑖𝐿 = 𝑓𝑖

𝑉 N

𝐾𝑖′ =

𝜙𝑖𝐿

𝜙𝑖𝑉

18

𝐼𝐹𝑇 = (∑ 𝑃𝜎𝑖𝑛𝑐𝑖=1 (

𝜌𝐿

𝑀𝑊𝐿𝑥𝑖 −

𝜌𝑉

𝑀𝑊𝑉𝑦𝑖))

4

(3.20)

𝜌𝐿

𝑀𝑊𝐿=

𝑃

𝑍𝐿𝑅𝑇 (3.21)

𝜌𝑉

𝑀𝑊𝑉=

𝑃

𝑍𝑉𝑅𝑇 (3.22)

Where 𝑃𝜎𝑖 is the parachor value of component i. 𝜌𝐿 and 𝜌𝑉 are the densities of the liquid

phase and vapor phase, respectively. 𝑀𝑊𝐿 and 𝑀𝑊𝑉 are the molecular weights of the

liquid and vapor phases, respectively. The units of IFT and density used here are mN/m

and mole/cm3, respectively.

3.4 Liquid Viscosity Calculation by Lohrenz Correlation

In this thesis, the liquid viscosity was determined by Lohrenz correlation (Lohrenz

et al. 1964). In this section, we present the Lohrenz correlation equations and calculation

procedures.

According to the Lohrenz correlation, the liquid viscosity is calculated by

𝜇 = 𝜇∗ + 𝜉𝑚−1 × [(0.1023 + 0.023364𝜌𝑃𝑟 + 0.058533𝜌𝑃𝑟

2 − 0.040758𝜌𝑃𝑟3 +

0.0093324𝜌𝑃𝑟4 )4 − 10−4] (3.23)

where 𝜇 is the fluid viscosity, 𝜇∗ is the mixture viscosity at the atmospheric pressure, 𝜉𝑚

is the mixture viscosity parameter, and 𝜌𝑃𝑟 is the reduced liquid density. 𝜇∗ can be

calculated by

𝜇∗ =∑ 𝑧𝑖𝜇𝑖

∗√𝑀𝑊𝑖𝑖

∑ 𝑧𝑖𝑖 √𝑀𝑊𝑖 (3.24)

19

where 𝑧𝑖 is the mole composition of component i in the mixture, 𝑀𝑊𝑖 is the molecular

weight of component i, and 𝜇𝑖∗ is the viscosity of component i at low pressure, which can

be calculated by

𝜇𝑖∗ =

17.78×10−5×(4.58𝑇𝑟𝑖−1.67)0.625

𝜉𝑖 (𝑖𝑓 𝑇𝑟𝑖 > 1.5) (3.25)

𝜇𝑖∗ =

34×10−5×𝑇𝑟𝑖0.94

𝜉𝑖 (𝑖𝑓 𝑇𝑟𝑖 ≤ 1.5) (3.26)

where 𝑇𝑟𝑖 is the reduced temperature for component i. The viscosity parameter of

component i, 𝜉𝑖 and the mixture viscosity parameter 𝜉𝑚 can be respectively calculated by

𝜉𝑖 =5.35×𝑇𝑐𝑖

1/6

√𝑀𝑊𝑖𝑃𝑐𝑖2/3 (3.27)

𝜉𝑚 =5.35×𝑇𝑝𝑐

1/6

√𝑀𝑊𝑚𝑃𝑝𝑐2/3 (3.28)

where 𝑇𝑝𝑐 is the pseudocritical temperature, 𝑃𝑝𝑐 is the pseudocritical pressure and 𝑀𝑊𝑚

is the liquid mixture molecular weight. The reduced density of the liquid mixture 𝜌𝑃𝑟 is

calculated by

𝜌𝑃𝑟 = (𝜌

𝑀𝑊𝑚) 𝑉𝑝𝑐 (3.29)

where 𝑉𝑝𝑐 is the mixture pseudocritical molar volume. All mixture pseudocritical properties

are calculated by the mixing rule,

𝑇𝑝𝑐 = ∑ 𝑧𝑖𝑇𝑐𝑖 (3.30)

𝑃𝑝𝑐 = ∑ 𝑧𝑖𝑃𝑐𝑖 (3.31)

𝑉𝑝𝑐 = ∑ 𝑧𝑖𝑉𝑐𝑖 (3.32)

20

𝑇𝑐𝑖, 𝑃𝑐𝑖 and 𝑉𝑐𝑖 are the critical temperature (°R), pressure (psi) and molar volume (ft3/lb-

mol) of component i.

21

CHAPTER 4

SIMULATION OF MEMBRANE FILTRATION IN HYDROCARBON SATURATED

NANOPOROUS MEDIA

This chapter describes the development of the membrane filtration model

motivated by the chemical osmosis theory presented in the first chapter. Due to filtration

of certain components, we expect that at equilibrium, pressure and compositional

differences shall exist in different parts of a nanoporous reservoir. Methods to calculate

pressure and compositional differences from single-solute and multi-solute filtration

efficiency are provided.

4.1 Hydrocarbon Filtration Model Development

As stated in Chapter 2, shale can act as a semi-permeable membrane due to

electrostatic exclusion and / or steric hindrance. In this thesis, we focus on a hydrocarbon-

saturated porous medium. Because most hydrocarbon molecules in oil reservoirs are

neutrally charged and do not possess strong polarity, we assume that the primary filtration

mechanism is steric hindrance.

Figure 4.1 Dual-pore filtration model.

22

As shown in Figure 4.1, we consider a dual-pore system filled with a hydrocarbon

mixture. In the figure, System I and II are reservoir pores saturated with hydrocarbon

fluids. The channel in the middle connecting the two pores is a collection of nano-sized

pore throats that can be considered as a semi-permeable membrane. As an analogy to

the aqueous solutions, the components with larger molecular diameters that are restricted

from passing through the pore throats, are referred to as the solutes, and the lighter

components are considered as the solvents.

4.2 Filtration Equilibrium and Filtration Pressure

We use a binary hydrocarbon mixture filtration case to illustrate the filtration

equilibrium and the filtration pressure. In this case, the hydrocarbon fluid consists of two

components, 1 and 2, of which 1 is the unrestricted component that has a smaller

molecular diameter (solvent) and can pass through the pore throats in both directions

freely, 2 is the component restricted because of steric hindrance (solute). If the membrane

is an ideal membrane, component 2 would be completely restricted. If the membrane is

a non-ideal membrane, component 2 would be hindered to some degree, and only a part

of component 2 can pass through the pore throats.

Initially, System I and II are assumed to be isolated with each other, and there is

no hydrocarbon transfer through the membrane. Each System has its own initial

composition, pore volume, pressure, temperature and number of moles. Now, let us keep

the temperature of the entire system equal and constant, and hold the volume of System

I and the pressure of System II constant, and allow hydrocarbons to transfer between the

two systems. Due to the concentration difference of the solute, which is component 2,

between the two Systems, the unrestricted component 1 will flow to the high-

23

concentration side, and the restricted component 2 will flow reversely if the membrane is

non-ideal, in an attempt to equalize the solute concentration. Eventually, the “escaping

tendency” or chemical potential of the unrestricted component 1 on the two sides would

develop a difference that counter-balances the pressure difference between the two sides,

and the entire system reaches its equilibrium. Here, for simplicity, we assume that the

pressure difference between Systems I and II does not generate any pressure-driven flux

of component 1 across the membrane. The transport of component 1 across the

membrane is strictly diffusive driven by the chemical potential of component 1. This

assumption is reasonable when the membrane is made up by extremely small nanopores.

When the pressure-driven flux of component 1 is nearly zero, it is also reasonable to

assume that the chemical potentials of component 1 in Systems I and II are equal.

Therefore, we do not need to evaluate the chemical potential driven flux and the pressure

driven flux, which requires the transport coefficients of the membrane, and the problem

can be modeled as a static equilibrium.

Figure 4.2 Equilibrium state of dual-pore filtration model.

24

Figure 4.2 shows the equilibrium state of the dual-pore filtration model. Because of

the initial concentration difference and membrane filtration, the solute concentrations of

each side will be different at the equilibrium state, making the hydrocarbon fluid of one

side heavier than that of the other side. At the equilibrium state, the pressure

difference, 𝑃𝐹 = 𝑃𝐼 − 𝑃𝐼𝐼, is defined as the filtration pressure. For an ideal membrane, the

filtration pressure calculated accordingly is the theoretical or ideal filtration pressure. For

a non-ideal membrane, the pressure obtained is the observed or realistic filtration

pressure.

As we have mentioned earlier, pressure driven flux across the membrane is

neglected so that the problem can be modeled as a static equilibrium. When the

membrane is ideal but the pressure-driven flux is non-negligible, the pressure difference

across the membrane should be lower than the case where the pressure–driven flux is

ignored. Phenomenologically, this situation is similar to the case where the pressure-

driven flux is zero, but the membrane is non-ideal.

4.3 Fugacity-Based Filtration Efficiency Calculation

In the previous study by Geren (2014), a fugacity-based filtration efficiency 𝜔𝑓 was

defined by Eq. (4.1) to quantitatively describe the ability of a nanoporous medium acting

as a semi-permeable membrane.

𝜔𝑓𝑖= 1 − (

𝑓𝐼𝐼,𝑖𝐿

𝑓𝐼,𝑖𝐿 ) (4.1)

In Eq. (4.1), 𝑓𝐼,𝑖𝐿 and 𝑓𝐼𝐼,𝑖

𝐿 are the fugacities of the restricted component i in the liquid phases

of the unfiltered and filtered parts, respectively. The superscript L represents the liquid

phase, I and II represent the unfiltered and filtered parts, respectively. The value of 𝜔𝑓 is

25

defined between zero, for a non-selective membrane, and one, for an ideal membrane.

However, this fugacity-based filtration efficiency 𝜔𝑓 is not only a property of the

membrane but also a function of the mixture, because the degree of chemical potential

mismatch of a particular component depends on the mixture that it is within.

In an attempt to establish filtration efficiencies that can be mainly regarded as a

property of the “membrane” – the nanoporous throats – to specific components, starting

from the chemical osmosis theory, we developed models and calculation procedures for

both single-solute and multi-solute filtration scenarios. The model equations and

calculation procedures are presented in the following section.

4.4 Single-Solute and Multi-Solute Filtration Efficiency Calculation

From Section 4.2, assume that we know both the theoretical filtration pressure

𝜋𝑖𝑑𝑒𝑎𝑙 and the observed filtration pressure 𝜋𝑟𝑒𝑎𝑙 for a binary hydrocarbon mixture, the

filtration efficiency can be calculated according to the definition and the equation termed

by Staverman (1952).

𝜎 = (𝑃𝐹,𝑟𝑒𝑎𝑙

𝑃𝐹,𝑖𝑑𝑒𝑎𝑙 )𝐽𝑣=0 (4.2)

Eq. (4.2) describes a case where the solute only contains a single component, which is

the restricted component 2 in a binary mixture. When the hydrocarbon fluid contains

multiple solute species, they should be lumped together. Eq. (4.2), then, describes the

overall filtration efficiency of the membrane to the lumped solute and does not represent

the filtration efficiency of the membrane to individual solute species.

As mentioned in Chapter 2, van ’t Hoff osmotic equation is valid for an ideal solution

with low solute concentration. Because the structures and properties of reservoir

hydrocarbon molecules are similar to each other, we can, as a starting point, treat the

26

hydrocarbon saturated reservoir fluid as an ideal solution. For light oils, because the

concentration of restricted heavy components is low, they satisfy the low-solute-

concentration requirement of van ’t Hoff osmotic equation. In an attempt to establish a

correlation between the overall filtration efficiency of a lumped solute and the filtration

efficiences of individual solute species for a multi-solute solution, we introduce a multi-

component filtration equation, which is adapted from van ’t Hoff osmotic equation to our

hydrocarbon saturated filtration system.

𝑃𝐹 = 𝑅𝑇 × ∑𝑛𝑖

𝑉𝑠

𝑛𝑐𝑖=1 (4.3)

Here, 𝑃𝐹 is the filtration pressure, R is the gas constant and T is the temperature. nc is

number of components, 𝑛𝑖 is the molar number of component i, and 𝑉𝑠 is the volume of

the solution. Because we assume that there is no dissociation and association of

hydrocarbon molecules in the modelled hydrocarbon mixture under the temperature and

pressure condition of this thesis, the dimensionless van ’t Hoff factor equals to one. Like

the van ’t Hoff osmotic equation, this filtration equation gives the absolute filtration

pressure of a solution when it is separated from the pure solvent by an ideal membrane.

From Eq. (4.3), when a membrane lies in the middle of two solutions with different

concentrations, the pressure difference across the membrane is given by

𝑃𝐹 = (𝑅𝑇 × ∑𝑛𝑖

𝑉𝑠

𝑛𝑐𝑖=1 )

ℎ𝑖𝑔ℎ− (𝑅𝑇 × ∑

𝑛𝑖

𝑉𝑠

𝑛𝑐𝑖=1 )

𝑙𝑜𝑤 (4.4)

where subscripts high and low represent high concentration and low concentration,

respectively.

27

For a single-solute solution stored in a dual-pore system with different concentrations on

each side, the single-component filtration efficiency of this single solute 𝜎𝑖∗ can be

calculated by using the filtration equation at the equilibrium state as below,

𝜎𝑖∗ =

(𝑛

𝑉𝑠𝑅𝑇)

𝐼,𝑟𝑒𝑎𝑙− (

𝑛

𝑉𝑠𝑅𝑇)

𝐼𝐼,𝑟𝑒𝑎𝑙

(𝑛

𝑉𝑠𝑅𝑇)

𝐼,𝑖𝑑𝑒𝑎𝑙− (

𝑛

𝑉𝑠𝑅𝑇)

𝐼𝐼,𝑖𝑑𝑒𝑎𝑙

(4.5)

where I and II represent System I and II respectively.

For a multi-solute solution stored in a dual-pore system with different

concentrations on each side, the filtration efficiency of the lumped solute 𝜎𝑙 and those for

individual solute 𝜎𝑖 can be calculated respectively by using the filtration equations at the

equilibrium state as below,

𝜎𝑙 =(𝑅𝑇×∑

𝑛𝑖𝑉𝑠

𝑛𝑐𝑖=1 )

𝐼,𝑟𝑒𝑎𝑙− (𝑅𝑇×∑

𝑛𝑖𝑉𝑠

𝑛𝑐𝑖=1 )

𝐼𝐼,𝑟𝑒𝑎𝑙

(𝑅𝑇×∑𝑛𝑖𝑉𝑠

𝑛𝑐𝑖=1 )

𝐼,𝑖𝑑𝑒𝑎𝑙− (𝑅𝑇×∑

𝑛𝑖𝑉𝑠

𝑛𝑐𝑖=1 )


(4.6)

𝜎𝑖 =(

𝑛𝑖𝑉𝑠

𝑅𝑇)𝐼,𝑟𝑒𝑎𝑙

− (𝑛𝑖𝑉𝑠

𝑅𝑇)𝐼𝐼,𝑟𝑒𝑎𝑙

(𝑛𝑖𝑉𝑠

𝑅𝑇)𝐼,𝑖𝑑𝑒𝑎𝑙

− (𝑛𝑖𝑉𝑠

𝑅𝑇)𝐼𝐼,𝑖𝑑𝑒𝑎𝑙

(4.7)

where I and II represent System I and II respectively.

It can be noticed that, for the multi-solute solution, as there are other solutes

present, the solution volume 𝑉𝑠 in Eq. (4.7) counts the volume of other solutes, in addition

to the volume of solute i. Under the ideal solution approximation and the assumption that

the total volume of the solutes is far less than the volume of the solvent, it can be

considered that 𝜎𝑖 = 𝜎𝑖∗, and Eq. (4.7) is valid for calculating this individual solute filtration

efficiency. However, for non-ideal solutions, and for solutions where the volume of the

solutes is non-negligible compared to the total solution volume, this approach may

generate deviations from the results calculated by definition (Eq. (4.2)).

28

From a theoretical point of view, it is always tempting to relate 𝜎𝑙 to 𝜎𝑖. It is easy to

show that

𝜎𝑙 =(∑

𝜎𝑖𝑛𝑖𝑉𝑠

𝑛𝑐𝑖=1 )

𝐼,𝑖𝑑𝑒𝑎𝑙− (∑

𝜎𝑖𝑛𝑖𝑉𝑠

𝑛𝑐𝑖=1 )


(∑𝑛𝑖𝑉𝑠

𝑛𝑐𝑖=1 )

𝐼,𝑖𝑑𝑒𝑎𝑙− (∑

𝑛𝑖𝑉𝑠

𝑛𝑐𝑖=1 )


(4.8)

which can be further simplified as

𝜎𝑙 =(∑ 𝜎𝑖𝑀𝑖

𝑛𝑐𝑖=1 )

𝐼,𝑖𝑑𝑒𝑎𝑙− (∑ 𝜎𝑖𝑀𝑖

𝑛𝑐𝑖=1 )


(∑ 𝑀𝑖𝑛𝑐𝑖=1 )

𝐼,𝑖𝑑𝑒𝑎𝑙− (∑ 𝑀𝑖

𝑛𝑐𝑖=1 )


(4.9)

where 𝑀𝑖 is the molarity of solute i in the ideal dilute solution.

Eq. (4.9) is a useful relation that connects the overall filtration efficiency 𝜎𝑙 to the

individual filtration efficiency 𝜎𝑖 evaluated in the same mixture.

In reality, the liquid mixture may possess non-idealness, and the volume of the

solutes is not always negligible compared to the total volume of the mixture. In our

calculations for 𝜎𝑙, starting from individually assigned 𝜎𝑖, we used PR-EOS to compute 𝑃𝐹;

this approach allows us to incorporate within the accuracy of the EOS non-idealness and

the volume of the solutes. In addition to simulating the pressure depletion process and

reporting the changes in fluid properties along the depletion path, we will also present a

comparison between 𝜎𝑙 calculated by definition (Eq. (4.2)) and that based on the modified

van ’t Hoff osmotic equation (Eq. (4.9)).

29

CHAPTER 5

SIMULATION OF PRESSURE DEPLETION OF A SINGLE-CELL RESERVOIR WITH

AN INTERNAL MEMBRANE

Usually, hydrocarbon fluids stored in conventional reservoirs are considered be

able to flow to production wells without compositional change when the pressure change

along the flow path does not demand a phase change. However, this notion may not apply

to tight-oil reservoirs, such as shale. As we stated before, the membrane properties of

these nanoporous reservoirs to hydrocarbon molecules may lead to non-uniform pressure

distribution within the reservoir and compositional variations even at equilibrium. As a

result, the corresponding reservoir fluids phase behavior and properties, such as density,

viscosity, and interfacial tension, may need to be re-characterized if the compositional

variation is significant.

To theoretically investigate the effect of membrane filtration of nanoporous

reservoirs on production, we simulated a constant-composition expansion of a light oil

confined in a porous medium with an internal membrane, the process of which is

analogous to the pressure depletion of a single-cell reservoir with internal filtration. As

illustrated in Figure 4.1, the fluid stored in the medium is divided into two parts: one part

that is already filtered and can flow to a production well without compositional change,

and another part that is unfiltered and can replenish the filtered fluids according to the

filtration efficiencies of the components to the nanoporous throats.

Simulation of pressure depletion consists of two primary stages: the initial-

equilibrium stage, which solves the initial state of the “reservoir” with membrane

30

properties before pressure depletion; and the post-initial equilibrium stage that calculates

the equilibrium states of the reservoir along the pressure depletion.

5.1 Establishing Initial-Equilibrium before Pressure Depletion

Figure 5.1 shows the unfiltered part and filtered part of the reservoir. The channels

connecting these two parts represent the semi-permeable membrane, which can be ideal

or non-ideal.

Figure 5.1 Two-part single-cell reservoir model: Initial-equilibrium before pressure depletion.

Similar to the filtration equilibrium calculation presented in Chapter 4, part I and

part II are initially unequilibrated with each other, and there is no hydrocarbon transfer

through the membrane. Each part has its initial pore volume, pressure, temperature and

number of moles. The initial composition of the unfiltered part is assumed heavier than

the filtered part. Then, we start the initial equilibration by keeping the pressure of filtered

part at a constant value, which is above the bubble point pressure of the fluid in the

filtered part, and allowing the hydrocarbon fluids in I and II exchange components across

the membrane. After the equilibrium is reached, we perform the single-phase equilibrium

31

Figure 5.2 Computational procedure of initial-equilibrium stage.

𝑉(𝑃𝐼 , 𝑇, 𝑛𝐼) = 𝑉𝐼_𝑖𝑛𝑖𝑡𝑖𝑎𝑙

Assume molar number of unrestricted component that transferred between two parts ∆𝑛𝑚𝑖

Assume ∆𝑃 between Part I and Part II and compute 𝑃𝐼 = 𝑃𝐼𝐼 + ∆𝑃

Update molar number of restricted component in Part I and II.

𝑛𝐼𝑟𝑖 = 𝑛𝐼

𝑟𝑖 + ∆𝑛𝑟𝑖 𝑛𝐼𝐼𝑟𝑖 = 𝑛𝐼𝐼

𝑟𝑖 − ∆𝑛𝑟𝑖

Assume molar number of restricted component that transferred between two parts ∆𝑛𝑟𝑖

START

Liquid phase equilibrium calculation for Part II At 𝑃𝐼𝐼 , 𝑇, 𝑛𝐼𝐼

Compute fugacities of unrestricted components in Part II

[𝑓𝐼𝐼1 , 𝑓𝐼𝐼

2, … , 𝑓𝐼𝐼𝑛𝑐𝑚]

Update molar number of unrestricted component in Part I and II.

𝑛𝐼𝑚𝑖 = 𝑛𝐼

𝑚𝑖 + ∆𝑛𝑚𝑖 𝑛𝐼𝐼𝑚𝑖 = 𝑛𝐼𝐼

𝑚𝑖 − ∆𝑛𝑚𝑖

Liquid phase equilibrium calculation for Part I At 𝑃𝐼 , 𝑇, 𝑛𝐼

Compute fugacities of unrestricted components in Part I

[𝑓𝐼1 , 𝑓𝐼

2 , … , 𝑓𝐼𝑛𝑐𝑚]

[𝑓𝐼1, … , 𝑓𝐼

𝑛𝑐𝑚] = [𝑓𝐼𝐼1, … , 𝑓𝐼𝐼

𝑛𝑐𝑚]

Y

o

Y

o

N

o

N

o

𝜎 = [𝜎𝑛𝑐𝑚+1, … , 𝜎𝑛𝑐𝑚+𝑛𝑐𝑟]

Compute filtration efficiency for every individual restricted component

∆𝑛𝑟𝑖 = 0 ∆𝑛𝑚𝑖 = 0

END

Y

o

Y

o

N

o

N

o

32

property calculations and filtration efficiency calculations to obtain the initial compositions

and properties of fluids in I and II for a given set of membrane efficiencies.

The known and unknown variables for the initial-equilibrium stage are listed in

Table 5.1. P is the pressure, V is the pore volume, T is the temperature, and n is the molar

number of every component. As shown in Table 5.1, there are 2𝑛𝑐 + 2 unknown variables,

where 𝑛𝑐 is the number of components.

Table 5.1 Known (√) and unknown (×) variables for the initial-equilibrium stage

Unfiltered Part Filtered Part

𝑃𝐼 × 𝑃𝐼𝐼 √

𝑉𝐼 √ 𝑉𝐼𝐼 ×

𝑇 √ 𝑇 √

[𝑛𝐼1, 𝑛𝐼

2, … , 𝑛𝐼𝑛𝑐] × [𝑛𝐼𝐼

1 , 𝑛𝐼𝐼2 , … , 𝑛𝐼𝐼

𝑛𝑐] ×

The following conditions constrain the equilibrium system: 1) The pore volume of

part I is a constant; 2) The fugacities of the unrestricted components in the unfiltered part

and the filtered part should be identical; 3) As we are simulating a closed system, the

number of moles for every component is conserved; 4) The filtration efficiencies of the

semi-permeable membrane for every individual restricted component are provided. Listed

below are the 2𝑛𝑐 + 2 equations needed to describe the state of equilibrium.

𝑉𝐼 = 𝑉(𝑃𝐼 , 𝑇, 𝑛𝐼) = 𝑉𝐼,𝑖𝑛𝑖𝑡𝑖𝑎𝑙 (5.1)

𝑉𝐼𝐼 = 𝑉(𝑃𝐼𝐼 , 𝑇, 𝑛𝐼𝐼) (5.2)

[𝑓𝐼1, 𝑓𝐼

2, … , 𝑓𝐼𝑛𝑐𝑚] = [𝑓𝐼𝐼

1, 𝑓𝐼𝐼2, … , 𝑓𝐼𝐼

𝑛𝑐𝑚] (5.3)

33

[𝑛𝐼1, 𝑛𝐼

2, … , 𝑛𝐼𝑛𝑐] + [𝑛𝐼𝐼

1 , 𝑛𝐼𝐼2 , … , 𝑛𝐼𝐼

𝑛𝑐] = [𝑛1, 𝑛2, … , 𝑛𝑛𝑐] (5.4)

𝜎 = [𝜎𝑛𝑐𝑚+1, 𝜎𝑛𝑐𝑚+2, … , 𝜎𝑛𝑐𝑚+𝑛𝑐𝑟] (5.5)

where 𝑛𝑐𝑚 and 𝑛𝑐𝑟 are the number of mobile (unrestricted) and restricted components,

respectively.

The computational procedure is shown in Figure 5.2 on page 31. Note that the

number of moles of components in the filtered part are updated every loop.

5.2 Computing Post-Initial Equilibriums during Pressure Depletion

After achieving the initial-equilibrium, we reduce the pressure of the filtered part,

which is assumed to be connected to a production well. Figure 5.3 shows the state of

phases for the entire reservoir when the pressure of the filtered part is reduced below the

bubble point of the filtered fluids. Fluids in the unfiltered part are still in the liquid phase,

and the filtered part contains liquid and vapor phases.

Figure 5.3 Two-part single cell reservoir model: post-initial equilibrium during pressure depletion.

To calculate a post-initial equilibrium state of the reservoir, we keep the pressure

of filtered part at a constant value, which is below the bubble point pressure of the fluids

34

Figure 5.4 Computational procedure of post-initial equilibrium stage.

𝑉(𝑃𝐼 , 𝑇, 𝑛𝐼) = 𝑉𝐼_𝑖𝑛𝑖𝑡𝑖𝑎𝑙

Assume molar number of unrestricted component that transferred between two parts ∆𝑛𝑚𝑖

Assume ∆𝑃 between Part I and Part II and compute 𝑃𝐼 = 𝑃𝐼𝐼 + ∆𝑃

Update molar number of restricted component in Part I and liquid phase of Part II.

𝑛𝐼𝑟𝑖 = 𝑛𝐼

𝑟𝑖 + ∆𝑛𝑟𝑖 𝑛𝐼𝐼,𝐿𝑟𝑖 = 𝑛𝐼𝐼,𝐿

𝑟𝑖 − ∆𝑛𝑟𝑖

Assume molar number of restricted component that transferred between two parts ∆𝑛𝑟𝑖

START

Vapor-Liquid two-phase equilibrium calculation for Part II At 𝑃𝐼𝐼 , 𝑇, 𝑛𝐼𝐼 = 𝑛𝐼𝐼,𝐿 + 𝑛𝐼𝐼,𝑉

Compute fugacities of every component in both liquid and vapor phase of Part II

[𝑓𝐼𝐼,𝐿1 , 𝑓𝐼𝐼,𝐿

2 , … , 𝑓𝐼𝐼,𝐿𝑛𝑐 ] = [𝑓𝐼𝐼,𝑉

1 , 𝑓𝐼𝐼,𝑉2 , … , 𝑓𝐼𝐼,𝑉

𝑛𝑐 ]

Update molar number of unrestricted component in Part I and liquid phase of Part II.

𝑛𝐼𝑚𝑖 = 𝑛𝐼

𝑚𝑖 + ∆𝑛𝑚𝑖 𝑛𝐼𝐼,𝐿𝑚𝑖 = 𝑛𝐼𝐼,𝐿

𝑚𝑖 − ∆𝑛𝑚𝑖

Liquid phase equilibrium calculation for Part I At 𝑃𝐼 , 𝑇, 𝑛𝐼

Compute fugacities of unrestricted components in Part I


2, … , 𝑓𝐼𝑛𝑐𝑚]

[𝑓𝐼1, … , 𝑓𝐼

𝑛𝑐𝑚] = [𝑓𝐼𝐼,𝐿1 , … , 𝑓𝐼𝐼,𝐿

𝑛𝑐𝑚]

N

o

N

o

Y

o

Y

o

𝜎 = [𝜎𝑛𝑐𝑚+1, … , 𝜎𝑛𝑐𝑚+𝑛𝑐𝑟]

Compute filtration efficiency for every individual restricted component

∆𝑛𝑟𝑖 = 0 ∆𝑛𝑚𝑖 = 0

END

Y

o

Y

o

N

o

N

o

35

in the filtered part, and allow hydrocarbons transfer across the membrane. After that, we

perform liquid-phase equilibrium property calculation, vapor-liquid two-phase equilibrium

calculation, and filtration efficiency calculation to find the equilibrium state.

The known and unknown variables for this calculation are listed in Table 5.2. P is

the pressure, V is the pore volume, T is the temperature, and 𝑛𝑖 is the molar number of

moles of every component. There are 3𝑛𝑐 + 2 unknown variables, where 𝑛𝑐 is the

number of components.

Table 5.2 Known (√) and unknown (×) variables for the post-initial equilibrium stage

Unfiltered Part Filtered Part

Liquid Phase Liquid Phase Vapor Phase

𝑃𝐼 × 𝑃𝐼𝐼 √ 𝑃𝐼𝐼 √

𝑉𝐼 √ 𝑉𝐼𝐼,𝐿 × 𝑉𝐼𝐼,𝑉 NA

𝑇𝐼 √ 𝑇𝐼𝐼 √ 𝑇𝐼𝐼 √

[𝑛𝐼1, … , 𝑛𝐼𝑛𝑐

𝑛𝑐 ] × [𝑛𝐼𝐼,𝐿1 , … , 𝑛𝐼𝐼,𝐿

𝑛𝑐 ] × [𝑛𝐼𝐼,𝑉1 , … , 𝑛𝐼𝐼,𝑉

𝑛𝑐 ] ×

The following conditions constrain the post-initial equilibrium states: 1) The pore

volume of part I is constant; 2) The fugacities of unrestricted components in the liquid

phase of filtered part, liquid phase of unfiltered part, and the vapor phase of filtered part

are equal; 3) the number of moles is conserved for every component; 4) The filtration

efficiencies of the semi-permeable membrane to every restricted component are provided.

Listed below are the 3𝑛𝑐 + 2 equations needed to solve the post-initial equilibrium state.

𝑉𝐼 = 𝑉(𝑃𝐼 , 𝑇𝐼 , 𝑛𝐼) = 𝑉𝐼,𝑖𝑛𝑖𝑡𝑖𝑎𝑙 (5.6)

36

𝑉𝐼𝐼,𝐿 = 𝑉(𝑃𝐼𝐼 , 𝑇𝐼𝐼 , 𝑛𝐼𝐼,𝐿) (5.7)


2, … , 𝑓𝐼𝑛𝑐𝑚] = [𝑓𝐼𝐼,𝐿

1 , 𝑓𝐼𝐼,𝐿2 , … , 𝑓𝐼𝐼,𝐿

𝑛𝑐𝑚] (5.8)

[𝑓𝐼𝐼,𝐿1 , 𝑓𝐼𝐼,𝐿

2 , … , 𝑓𝐼𝐼,𝐿𝑛𝑐 ] = [𝑓𝐼𝐼,𝑉

1 , 𝑓𝐼𝐼,𝑉2 , … , 𝑓𝐼𝐼,𝑉

𝑛𝑐 ] (5.9)

[𝑛𝐼1, … , 𝑛𝐼

𝑛𝑐] + [𝑛𝐼𝐼,𝐿1 , … , 𝑛𝐼𝐼,𝐿

𝑛𝑐 ] + [𝑛𝐼𝐼,𝑉1 , … , 𝑛𝐼𝐼,𝑉

𝑛𝑐 ] = [𝑛1, … , 𝑛𝑛𝑐 ] (5.10)

𝜎 = [𝜎𝑛𝑐𝑚+1, 𝜎𝑛𝑐𝑚+2, … , 𝜎𝑛𝑐𝑚+𝑛𝑐𝑟] (5.11)

where 𝑛𝑐𝑚 and 𝑛𝑐𝑟 are the number of mobile (unrestricted) and restricted components,

respectively.

The computational procedure is shown in Figure 5.4 on page 34. Note that the

number of moles of components in the unfiltered part are updated every loop.

37

CHAPTER 6

RESULTS AND DISCUSSIONS

In this chapter, we present simulation results of pressure depletion processes of a

light oil confined in porous media with internal membranes. The light oil used in these

simulations consists of nC4, nC10, nC16 and C24. Among all the components, nC16 and C24

are the components restricted from moving through the pore throats, and therefore,

subjected to membrane filtration. The other components can move freely through the pore

throats without hindrance. Table 6.1 shows the thermodynamic parameters of the

components, including critical properties, acentric factors, molecular weights and binary

interaction parameters. Table 6.2 displays the initial state parameters of the unfiltered

and filtered parts, including pressure, temperature, and composition of each part, etc.

Table 6.1 Thermodynamic parameters of components in the light oil

Table 6.2 Initial state parameters

Properties Unfiltered Part I Filtered Part II

Pressure (psi) 5000 5000

Temperature (°K) 360.9 360.9

Liquid phase volume (10-25 m3) 5.000 5.000

Composition (nC4-nC10-nC16-C24, mol%)

10.0 10.0 30.0 50.0 30.0 30.0 20.0 20.0

Parameters nC4 nC10 nC16 C24

Tc (⁰K) 425.2 617.6 717 823.2

Pc (psi) 551.1 305.7 205.7 181.9

Vc (m3/kmol) 0.255 0.603 0.956 1.17

ω 0.193 0.49 0.742 0.94

MW (g/mol) 58.124 142.286 226.448 324

δij

nC4 0 0.012228 0.028461 0.037552

nC10 0.012228 0 0.00353 0.007281

nC16 0.028461 0.00353 0 0.00068

C24 0.037552 0.007281 0.00068 0

38

Table 6.3 presents the results of the initial-equilibrium stage before pressure

depletion. The filtration efficiencies of the membrane to nC16 and C24 are 0.35 and 0.55,

respectively. The viscosities were calculated using the Lohrenz correlation (Lohrenz et al.

1964). The Lohrenz viscosity correlation has been presented in Section 3.4.

Table 6.3 The initial-equilibrium state before pressure depletion



Temperature (⁰K) 360.9 360.9



18.22 14.90 26.42 40.46 24.76 27.58 22.17 25.49

Number of moles (nC4-nC10-nC16-C24, 10-22 mol)

2.825 2.310 4.098 6.275 4.524 5.039 4.049 4.656

Apparent molecular weight 222.7 186.4

Molar volume (m3/kmol) 0.322 0.275

Viscosity (cp) 0.226 0.237

Density (kg/m3) 690.8 677.3

After achieving the initial-equilibrium state before pressure depletion, we reduced

the pressure of the unfiltered part, and performed equilibrium and filtration calculations to

solve the corresponding post-initial equilibrium state. To illustrate the effect of membrane

filtration efficiencies, we simulated pressure depletion processes with different

membranes:

Case I: Ideal membrane, 𝜎 = [1, 1].

Case II: Non-ideal membrane, 𝜎 = [0.35, 0.55].

39

Case II has the same membrane efficiencies as the initial-equilibrium stage. Case I, on

the other hand, uses the ideal membrane to explore the effect of different time scales

between initial-equilibrium and post-initial equilibrium. The initial-equilibrium stage usually

involves geological time scales. The post-initial equilibriums, on the other hand, simulates

the states of the reservoir during the much more rapid depletion. We, therefore, anticipate

that the membrane would appear to be ideal in this stage compared to that for the initial-

equilibrium, because of the long time needed for the heavy components to transport

through the membrane.

Additionally, to illustrate the effect of membrane properties, we simulated a

pressure depletion process starting from the overall composition of the initial state (cf.

Table 6.2) with a non-selective membrane:

Case III: Non-selective membrane, 𝜎 = [0, 0].

For Case III, during the initial-equilibrium stage and the post-initial equilibrium stage, the

reservoir is assumed as a conventional reservoir, which has no membrane properties.

Table 6.4, 6.5, and 6.6 show the state of phases as well as fluid properties at the

post-initial equilibrium state for the above three cases when depletion pressure is 30 psi,

respectively. The overall filtration efficiency for every case is also calculated. Table 6.7

shows the three cases’ overall and individual filtration efficiencies.

We also simulated a pressure depletion process starting from an initial-equilibrium

state with a filtration efficiency of [0.75, 0.9], and post-equilibrium states with filtration

efficiencies of [0.75, 0.9] and [1, 1], respectively. Those simulation results are presented

in Appendix B.

40

Table 6.4 Post-initial equilibrium state when the pressure of the filtered part is reduced to 30 psi. Ideal membrane σ = [1, 1]






1.751 2.386 4.098 6.275 5.598 4.963 4.049 4.656


12.07 16.44 28.24 43.25 29.06 25.76 21.02 24.16


Liquid phase molar number (10-22 mol) 1.751 2.386 4.098 6.275 2.697 4.932 4.049 4.656

Liquid phase composition (mol%) 12.07 16.44 28.24 43.25 16.51 30.20 24.79 28.50

Liquid phase molecular weight 234.5 201.0

Vapor phase molar number (10-22 mol) -- 2.900 0.031 2.89E-04 1.75E-06

Vapor phase composition (mol%) -- 98.95 1.04 0.01 5.96E-05

Vapor phase molecular weight -- 59.02

Liquid phase fraction (mol%) 100 84.8

Molar volume of Liquid phase (m3/kmol) 0.345 0.304

Viscosity of liquid phase (cp) 0.197 0.192

Interfacial tension (mN/m) -- 11.087

Density of liquid phase (kg/m3) 680.5 661.0

41

Table 6.5 Post-initial equilibrium state when the pressure of the filtered part is reduced to 30 psi. Non-ideal membrane σ = [0.35, 0.55]






2.090 2.957 4.067 5.887 5.258 4.392 4.079 5.044


13.93 19.71 27.11 39.24 28.01 23.39 21.73 26.87













42

Table 6.6 Post-initial equilibrium state when the pressure of the filtered part is reduced to 30 psi. Non-selective membrane, σ = [0, 0]

Properties Entire System

Pressure (psi) 30

Temperature (⁰K) 360.9

Composition (nC4-nC10-nC16-C24, mol%) 21.76 21.76 24.12 32.36

Apparent molecular weight 203.1

Liquid phase composition (mol%) 16.33 23.23 25.81 34.63

Liquid phase apparent molecular weight 213.2

Vapor phase composition (mol%) 99.18 0.81 0.01 7.118E-05

Vapor phase apparent molecular weight 58.8

Liquid phase fraction (mol%) 93

Molar volume of liquid phase (m3/kmol) 0.320

Viscosity of liquid phase (cp) 0.190

Interfacial tension (mN/m) 10.829

Density of liquid phase (kg/m3) 666.2

43

Table 6.7 Overall and individual filtration efficiencies for every case

Case I Case II Case III

Individual Filtration Efficiency σ nC4 nC10 nC16 C24 nC4 nC10 nC16 C24 nC4 nC10 nC16 C24

0.0 0.0 1.0 1.0 0.0 0.0 0.3590 0.5461 0.0 0.0 0.0 0.0

Overall Filtration Efficiency,

from 𝜎 = (𝑃𝐹,𝑟𝑒𝑎𝑙

𝑃𝐹,𝑖𝑑𝑒𝑎𝑙 )𝐽𝑣=0

1.0 0.516 0.0

Overall Filtration Efficiency, from Eq. (4.9) 1.0 0.544 0.0

44

For all the results presented above, from Table 6.4 to Table 6.7, Case I and II are

calculated from the initial-equilibrium state established with a filtration efficiency of [0.35,

0.55], Case III is calculated from the initial state with a non-selective membrane.

From Table 6.7, it is noted that when individual filtration efficiencies are unity, the

overall filtration efficiency is also unity. When the individual filtration efficiency of restricted

components (nC16, C24) for a non-ideal membrane case is set as [0.35, 0.55], the overall

filtration efficiency calculated using Eq. (4.2) is 0.516. The overall filtration efficiency

calculated by Eq. (4.9) equals to 0.544, which is not far from 0.516.

The filtration efficiency calculated by definition or PR-EOS should be more accurate.

However, it is much easier to calculate the overall filtration efficiency by using the modified

van ’t Hoff equation, Eq. (4.9).

Table 6.8 and 6.9 present the pressures of unfiltered and filtered parts at different

depletion pressures for nanoporous media with an ideal membrane or a non-ideal

membrane, respectively. From Table 6.8 and 6.9, it can be noted that as the depletion

pressure decreases, the pressures of unfiltered and filtered parts also decrease. The

filtration pressure, which is the difference between the pressure of unfiltered part and

filtered part, for each membrane case is plotted as a function of depletion pressure in

Figure 6.1. From Figure 6.1, it can be noted that the less the filtration efficiency, the lower

the filtration pressure between the unfiltered part and filtered part.

For a nanoporous medium with an ideal membrane, there is no transfer of restricted

components (nC16, C24) through the membrane. For a nanoporous medium with a non-

ideal membrane, restricted components (nC16, C24) can still move from the high-

concentration side (unfiltered part) to the low-concentration side (filtered part). Due to

45

membrane filtration, during pressure depletion, the hydrocarbon mixture in the unfiltered

part becomes heavier, and the hydrocarbon mixture in the filtered part becomes lighter.

The higher the filtration efficiency, the more significant is this trend. Associated with the

vaporization of the hydrocarbon mixture in the filtered part, and, for the case of a non-

ideal membrane, the amount of restricted components (nC16, C24) moving through the

membrane, the liquid phase in the filtered part turns heavier over time. These

expectations have all been verified by our simulation results. We calculated the molecule

weights of the fluid mixtures, which consist of both gas and liquid phases, in the filtered

part and unfiltered part and the molecule weight of the liquid phase in the filtered part at

several other depletion pressures (unfiltered part pressure) ranging from 80 psi to 20 psi

for an ideal and a non-ideal membrane case, respectively. These molecule weights are

presented in Table 6.10 and 6.11. Also, we plotted these fluid molecular weights for each

case as a function of depletion pressure in Figure 6.2 and 6.3, separately. When there is

no internal filtration, the molecule weights of the fluid should remain as a constant (203)

because the process is a constant-composition expansion process. From Figure 6.2 and

6.3, it can be noted that, in both ideal and non-ideal case, the molecule weight of the

mixture in the filtered part decreases as depletion pressure decreases, and the molecule

weight of the mixture in the unfiltered part increases as depletion pressure decreases.

Besides, the difference between the molecular weights of the mixture in the filtered and

unfiltered parts for an ideal membrane case is higher than that for a non-ideal membrane

case. The green line, which represents the molecule weight of the liquid in the filtered

part, goes up as depletion pressure decreases, indicating the liquid phase in the filtered

part turns heavier over time.

46

Table 6.8 Pressure of unfiltered/filtered part at different depletion pressures, ideal membrane case

Depletion Pressure (psi) 80 60 55 50 45 40 35 30 25 20

Pressure of Unfiltered Part (psi) 1909 1637 1565 1491 1415 1339 1257 1173 1087 1000

Pressure of Filtered Part (psi) 80 60 55 50 45 40 35 30 25 20

Filtration Pressure (psi) 1829 1577 1510 1441 1370 1299 1222 1143 1062 980

Table 6.9 Pressure of unfiltered/filtered part at different depletion pressures, non-ideal membrane case


Pressure of Unfiltered Part (psi) 1324 1019 928 857 700 677 670 620 586 551

Pressure of Filtered Part (psi) 80 60 55 50 45 40 35 30 25 20

Filtration Pressure (psi) 1244 959 873 807 655 637 635 590 561 531

47

Figure 6.1 Pressure difference between filtered and unfiltered part at different depletion pressures for different cases.

400

600

800

1000

1200

1400

1600

1800

2000

1525354555657585

Filt

ratio

n p

ressu

re, p

si

Depletion pressure, psi

Non-ideal membrane Ideal membrane

48

Table 6.10 Molecule weight of fluid in unfiltered/filtered part at different depletion pressures, ideal membrane case

Depletion Pressure (psi) 60 55 50 45 40 35 30 25 20

Fluid Mixture in Unfiltered Part 221 223 225 227 230 232 234 237 240

Fluid Mixture in Filtered Part 188 187 185 186 182 181 179 178 176

Liquid in Filtered Part 188 187 185 185 190 196 201 206 212

Table 6.11 Molecule weight of fluid in unfiltered/filtered part at different depletion pressures, non-ideal membrane case

Depletion Pressure (psi) 60 55 50 45 40 35 30 25 20

Fluid Mixture in Unfiltered Part 208 210 212 212 217 221 225 228 233

Fluid Mixture in Filtered Part 199 197 195 195 191 188 186 183 181

Liquid in Filtered Part 199 197 195 195 196 201 206 212 217

49

Figure 6.2 Molecule weight of fluid in unfiltered/filtered part at different depletion pressures, ideal membrane.

170

180

190

200

210

220

230

240

250

1525354555657585

Mo

lecu

le w

eig

ht


Unfiltered Part Filtered Part Liquid Phase Filtered Part

50

Figure 6.3 Molecule weight of fluid in unfiltered/filtered part at different depletion pressures, non-ideal membrane.

170

180

190

200

210

220

230

240

250

1520253035404550556065

Mo

lecu

le w

eig

ht


Unfiltered Part Filtered Part Liquid Phase Filtered Part

51

Table 6.12 and Figure 6.4 present the vapor phase molar fraction in the filtered part

of nanoporous media with different membranes at different depletion pressures. For every

case, as depletion pressure decreases, liquid phase keeps vaporizing and the vapor

phase molar fraction increases. Also, it can be noted that as depletion pressure

decreases, the fluid mixture in the filtered part for an ideal membrane case is the first one

to vaporize, and the fluid mixture for a non-selective membrane case is the last one to

vaporize. This result indicates that the higher the filtration efficiency, the higher the bubble

point pressure of the fluid mixture in the filtered part of a nanoporous medium.

To verify the implemented phase equilibrium and phase property calculations, we

obtained another set of vapor phase molar fraction data by inputting the compositions of

the fluid mixture in the filtered part at different depletion pressures for every case into

WinProp, and performing the QNSS/Newton-based two-phase flash calculation. The

validation results are presented in Table 6.13 and plotted as a function of depletion

pressure in Figure 6.5. The fact that they are nearly close to Table 6.12 and Figure 6.4

shows the correctness of our simulation.

Table 6.14 and Figure 6.6 present the viscosities of the liquid in the unfiltered and

filtered parts at different depletion pressures for different cases. From Figure 6.6, it can

be noted that as depletion pressure decreases, the viscosities of the liquid in both of the

filtered part and unfiltered part decrease. By comparing the liquid viscosities in the filtered

part and unfiltered part for an ideal case or a non-ideal case, we find that the viscosity of

the filtered liquid is less than that of the unfiltered liquid. As there is no internal filtration in

a non-selective membrane case, the viscosities of the liquid in the filtered part and

unfiltered part in a nanoporous medium without membrane are same. These simulation

52

results were also validated by WinProp. We calculated the liquid viscosity in each part for

different cases at different pressures. The results from WinProp are presented in Table

6.15 and plotted in Figure 6.7 as a function of depletion pressure. By comparing our

simulation results with the results from Winprop, we confirmed that Lohrenz correlation

for viscosity calculation has been implemented correctly.

Table 6.16 and Figure 6.8 present the interfacial tensions between the vapor and

liquid phase in the unfiltered part at different depletion pressures for different cases. From

Figure 6.8, it can be noted that IFT increases as the depletion pressure decreases. This

trend is similar to the experimental data for the interfacial tensions between equilibrium

oil and gas phases of several reservoir fluids, presented in Figure 6.9. In addition, from

Figure 6.9, as pressure decreases and approaches 0 psi, measured IFT increases rapidly,

and finally equals approximately 20 dyne/cm, or 20 mN/m (dyne/cm = mN/m). As we only

picked four hydrocarbon components, for simplification, to represent the light oil in our

model, our simulated results are less than those measured values but the sensitivities on

the pressure are similar.

Table 6.17 presents the density of the liquid and vapor phase in the filtered part at

different depletion pressures for different cases. Figure 6.10 and Figure 6.11 present the

density of the liquid and vapor phases in the filtered part, respectively. From Figure 6.10,

it can be noted that when pressure is above the bubble point pressure of the filtered fluid,

the liquid density decreases as pressure decreases, when the pressure is below the

bubble point pressure, the density increases as pressure decreases, due to the

vaporization of the light components in the filtered part. It can also be noted that the

density of the liquid in the filtered part of a nanoporous medium with higher filtration

53

efficiency is higher than that with a lower filtration efficiency. From Figure 6.11, it can be noted that as depletion pressure

decreases, due to the vaporization of the light components, the density of the vapor phase in the filtered part decreases.

Table 6.12 Vapor phase molar fraction in filtered part at different depletion pressures for different cases

Depletion pressure (psi) 50 45 40 35 30 25 20

Ideal membrane case 0 0.98 5.92 10.72 15.22 19.36 23.34

Non-ideal membrane case 0 0 3.62 8.94 14.01 18.65 22.95

Non-selective membrane case (Reference) 0 0 0 3.38 6.55 9.54 12.38

Table 6.13 Vapor phase molar fraction in filtered part at different depletion pressures for different cases, validation results from WinProp

Depletion pressure (psi) 50 45 40 35 30 25 20

Ideal membrane case 0 1.13 6.15 10.84 16.70 19.44 23.41

Non-ideal membrane case 0 0 3.73 9.19 14.09 18.70 22.99

Non-selective membrane case (Reference) 0 0 0.18 3.52 6.66 9.63 12.44

54

Figure 6.4 Vapor phase molar fraction in filtered part at different depletion pressures for different cases.

0

5

10

15

20

25

2025303540455055

Va

po

r p

ha

se

mo

lar

fra

ctio

n, %


Ideal membrane Non-ideal membrane Reference system

55

Figure 6.5 Vapor phase molar fraction in filtered part at different depletion pressures for different cases, validation results from WinProp.

0

5

10

15

20

25

55 50 45 40 35 30 25 20

Va

po

r p

ha

se

mo

lar

fra

ctio

n, %


Reference system Non-ideal membrane Ideal membrane

56

Table 6.14 Viscosity of liquid in unfiltered/filtered part at different depletion pressures for different cases


Ideal membrane case

Unfiltered Part Liquid Viscosity (cp)

0.2096 0.2054 0.2042 0.2029 0.2015 0.2001 0.1985 0.1967 0.1949 0.1928

Filtered Part Liquid Viscosity (cp)

0.1942 0.1940 0.1939 0.1938 0.1936 0.1934 0.1929 0.1921 0.1911 0.1897

Non-ideal membrane case


0.2056 0.2017 0.2004 0.1993 0.1972 0.1964 0.1953 0.1937 0.1923 0.1903


0.1929 0.1927 0.1928 0.1928 0.1925 0.1924 0.1919 0.1910 0.1899 0.1885

Non-selective membrane case

(Reference) Liquid Viscosity (cp) 0.1918 0.1915 0.1915 0.1914 0.1914 0.1913 0.1906 0.1896 0.1884 0.1869

57

Table 6.15 Viscosity of liquid in unfiltered/filtered part at different depletion pressures for different cases, validation results from WinProp


Ideal membrane case


0.2061 0.2020 0.2008 0.1995 0.1982 0.1967 0.1952 0.1935 0.1918 0.1898


0.1937 0.1936 0.1935 0.1933 0.1931 0.1930 0.1924 0.1916 0.1905 0.1892

Non-ideal membrane case


0.2038 0.1999 0.1987 0.1975 0.1955 0.1947 0.1935 0.1920 0.1904 0.1886


0.1923 0.1923 0.1923 0.1924 0.1922 0.1919 0.1912 0.1903 0.1893 0.1879

Non-selective membrane case

(Reference) Liquid Viscosity (cp) 0.1913 0.1911 0.1910 0.19097 0.1909 0.1908 0.1900 0.1890 0.1878 0.1863

58

Figure 6.6 Viscosity of liquid in unfiltered/filtered part at different depletion pressures for different cases.

0.185

0.19

0.195

0.2

0.205

0.21

0.215

102030405060708090

Liq

uid

vis

co

sity,

cp


Unfiltered, Idealmembrane

Unfiltered, Non-ideal membrane

Filtered, Idealmembrane

Filtered, Non-idealmembrane

Reference System

59

Figure 6.7 Viscosity of liquid in unfiltered/filtered part at different depletion pressures for different cases, validation results from WinProp.

0.185

0.19

0.195

0.2

0.205

0.21

102030405060708090

Liq

uid

Vis

co

sity,

cp


Unfiltered, Idealmembrane

Unfiltered, Non-ideal membrane

Filtered, Idealmembrane

Filtered, Non-idealmembrane

Reference System

60

Table 6.16 Vapor/Liquid interfacial tensions in filtered part at different depletion pressures for different cases

Depletion pressure (psi) 45 40 35 30 25 20

Ideal membrane, IFT (mN/m) 10.8781 10.9525 11.0214 11.0867 11.1501 11.2108

Non-ideal membrane, IFT (mN/m) -- 10.8246 10.9040 10.9669 11.0362 11.0976

Non-selective membrane (Reference), IFT (mN/m) -- -- 10.7576 10.8294 10.8994 10.9675

61

Figure 6.8 Vapor/Liquid interfacial tensions in filtered part at different depletion pressures for different cases.

10.70

10.75

10.80

10.85

10.90

10.95

11.00

11.05

11.10

11.15

11.20

11.25

1520253035404550

Inte

rfa

cia

l te

nsio

n, m

N/m


Ideal membrane Non-ideal membrane Reference System

62

Figure 6.9 Interfacial tension vs. Pressure for various reservoir oils. (Firoozabadi et al. 1988)

63

Table 6.17 Density of fluids in filtered part at different depletion pressures for different cases

Depletion pressure (psi) 80 60 55 50 45 40 35 30 25 20

Ideal membrane

Density (kg/m3)

Liquid 659.1 657.0 656.5 655.9 655.8 657.5 659.3 661.0 662.5 664.1

Vapor -- -- -- -- 6.41 5.67 4.94 4.22 3.50 2.80

difference -- -- -- -- 649.3 651.8 654.3 656.8 659.0 661.3

Non-ideal membrane

Density (kg/m3)

Liquid 663.6 661.2 660.7 659.9 660.0 660.3 661.7 663.4 664.9 666.3

Vapor -- -- -- -- -- 5.66 4.93 4.21 3.50 2.79

difference -- -- -- -- -- 654.6 656.8 659.2 661.4 663.5

Non-selective membrane

(Reference)

Density (kg/m3)

Liquid 663.6 663.5 663.5 663.4 663.4 663.4 664.8 666.2 667.6 668.9

Vapor -- -- -- -- -- -- 4.92 4.20 3.49 2.79

difference -- -- -- -- -- -- 659.9 662.0 664.1 666.1

64

Figure 6.10 Density of liquid phase in filtered part at different depletion pressures for different cases.

654

656

658

660

662

664

666

668

670

10203040506070

Liq

uid

ph

ase

de

nsity,

kg

/m3



65

Figure 6.11 Density of vapor phase in filtered part at different depletion pressures for different cases.

0

1

2

3

4

5

6

7

1520253035404550

Va

po

r p

ha

se

de

nsity,

kg

/m3



66

CHAPTER 7

SUMMARY, CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE WORK

This chapter presents the summary of our work and the conclusions of this

simulation study. The future work of the study on membrane effect is discussed at the

end of this chapter.

7.1 Summary

In this theoretical study, we investigated the effect of membrane properties of a

porous medium on the depletion process of a tight-light oil reservoir through developing

and then using a numerical model to simulate the phase behavior, fluid properties and

transfer of reservoir fluids during the depletion process. We established filtration efficiency

models and filtration equations using the chemical osmosis theory as an analog. They

can be applied to quantitatively predict the effect of porous media membrane properties

and filtration on reservoir fluids phase behavior.

7.2 Conclusions

By simulating a pressure depletion process for a porous medium with internal

membrane, we find that, as the pressure of the filtered part decreases, lighter components

in the filtered part (most of which are unrestricted components) vaporize into a gas phase,

increasing the molarity of the restricted components in the liquid phase of the filtered part.

In porous medium without membrane properties, components can move freely to

establish thermodynamic equilibrium across the entire medium after depletion. In contrast,

hydrocarbon components in a porous medium with an internal membrane can only move

through the membrane according to their respective filtration efficiencies, and filtration

pressures during the depletion. The selectivity of the membrane leads to a pressure

67

difference between the unfiltered and filtered parts of the porous medium as well as a

significant compositional difference. Membrane filtration makes the produced

hydrocarbon mixture lighter and traps the heavier components (most of which are

restricted components) in the reservoir. By performing equilibrium and filtration

calculations within a porous medium after pressure depletion, we obtained fluid saturation

distributions in the porous medium and the compositions of the produced and trapped

fluids. Also, we calculated the properties of the produced and trapped fluids, such as the

bubble point pressures, liquid viscosities, interfacial tensions between the vapor and liquid

phases, and fluids densities.

These findings and results can help us better understand the effects of internal

filtration in a nanoporous reservoir may have on the phase behavior and properties of

reservoir fluids during a pressure depletion, and may provide us new ideas to carry out

EOR in tight oil reservoirs.

7.3 Recommendations for Future Work

For a multi-solute solution, the overall filtration efficiency calculated by the modified

van ’t Hoff equation, Eq. (4.9) slightly deviates from the overall filtration efficiency

calculated by definition, Eq. (4.2). It is considered that the deviation is mainly caused by

the non-negligible volume of the solutes in the non-ideal solution. In the future, we hope

to be able to solve this issue by constructing a correlation between these two approaches

to save the effort spent in calculating the filtration efficiency.

In this preliminary model, we used four hydrocarbon components to represent a

light oil for simplification. Simulations of depletion for porous media with internal filtration

using various hydrocarbon components should be conducted in the future to better

68

understand the membrane effect on the phase behavior of reservoir fluids. Additionally,

we did not consider and incorporate capillary pressure in the iterative calculation

procedure of our model. This may bring deviations to our simulation results.

In this thesis, we introduce the chemical osmosis theory and for the first time

connect this theory with the membrane filtration process of hydrocarbons. Also, we solved

the modified osmotic equations to quantitatively describe the hydrocarbon transfer during

the filtration. Through simulating a pressure depletion process for a porous medium with

internal filtration, we obtain potentially experimentally verifiable predictions that could be

used to prove the effect of membrane filtration. We find that the membrane filtration can

make the produced hydrocarbon mixture lighter, and traps the heavier components in the

reservoir. In addition, the membrane can generate a pressure difference between the

unfiltered and filtered parts. These findings can be potentially examined by experiments.

However, because we did not consider all possible interactions between the reservoir

fluids and the nanoporous reservoir rocks in this simulation, the simulated fluids

compositions, and related fluids properties may deviate from the results obtained from

experiments. We need to perform molecule simulations to incorporate the more faithfully

rock-fluid interactions during the depletion. At this time, there is no experimental evidence

to support or disqualify these predictions. In the future, we hope to be able to obtain

reliable data from experiments or molecular simulations to verify the general trends during

the depletion of a porous medium with internal filtration, and validate and improve our

model and its predictions.

69

LIST OF SYMBOLS

a= Peng-Robinson equation constant

𝑎𝑖= Peng-Robinson equation constant of component i

𝑎𝑖𝑗= Peng-Robinson equation constant

A= Peng-Robinson equation constant

b= Peng-Robinson equation constant

𝑏𝑖= Peng-Robinson equation constant of component i

B= Peng-Robinson equation constant

𝑓𝑖𝐿= Fugacity of component i in the liquid phase, psi

𝑓𝑖𝑉= Fugacity of component i in the vapor phase, psi

𝑓𝐼𝑖= Fugacity of component i in the unfiltered part, psi

𝑓𝐼𝐼𝑖 = Fugacity of component i in the filtered part, psi

𝑓𝐼𝐼,𝐿𝑖 = Fugacity of component i in the liquid phase of the filtered part, psi

𝑓𝐼𝐼,𝑉𝑖 = Fugacity of component i in the vapor phase of the filtered part, psi

𝑖= Dimensionless van ’t Hoff factor

𝐽𝑣= Net fluid flux through the membrane

k= Pure component parameter in Peng-Robinson equation

𝑘−1= Effective thickness of electrical double layer

Ki= Distribution coefficient or K factor of component i

𝐾𝑖′= Updated distribution coefficient or K factor of component i

nc= Number of components

𝑛𝑐𝑚= Number of unrestricted components

𝑛𝑐𝑟= Number of restricted components

70

𝑛𝑖= Molar number of component i, mol

𝑛𝐼= Number of moles in the unfiltered part, mol

𝑛𝐼𝑖= Molar number of component i in the unfiltered part, mol

𝑛𝐼𝑚𝑖= Molar number of unrestricted component i in the unfiltered part, mol

𝑛𝐼𝑟𝑖= Molar number of restricted component i in the unfiltered part, mol

𝑛𝐼𝐼= Number of moles in the filtered part, mol

𝑛𝐼𝐼𝑖 = Molar number of component i in the filtered part, mol

𝑛𝐼𝐼𝑚𝑖= Molar numberof unrestricted component i in filtered part, mol

𝑛𝐼𝑟𝑖= Molar number of restricted component i in the filtered part, mol

𝑛𝐼𝐼,𝐿𝑖 = Molar number of component i in the liquid phase of filtered part, mol

𝑛𝐼𝐼,𝑉𝑖 = Molar number of component i in the vapor phase of filtered part, mol

𝑁𝑜= Liquid phase molar fraction, fraction

M= Molarity, lb−mole

ft3

MW= Molecular weight, g/mol

𝑀𝑊𝑖= Molecular weight of component i, lbm/lb-mol

𝑀𝑊𝑚=Molecular weight of mixture, lbm/lb-mol

𝑃= Pressure, psi

𝑃𝑐= Critical pressure, psi

𝑃𝑐𝑖= Critical pressure of component i, psi

𝑃𝐹= Filtration pressure, psi

𝑃𝐹,𝑟𝑒𝑎𝑙=Observed or realistic filtration osmotic pressure, psi

𝑃𝐹,𝑖𝑑𝑒𝑎𝑙=Theoretical filtration pressure, psi

71

𝑃𝑝𝑐= Pseudocritical pressure, psi

𝑃𝑟= Reduced pressure

𝑃𝜎𝑖= Parachor value of component i

𝑃𝐼= Pressure of unfiltered part, psi

𝑃𝐼𝐼= Pressure of filtered part, psi

R= Gas constant, ft3psi

lb−mole R

Ri= Ratio between the fugacity of component i in the liquid and vapor phase

T= Temperature, ⁰K

Tc = Critical temperature, ⁰K

𝑇𝑐𝑖= Critical temperature of component i, °R

𝑇𝑝𝑐= Pseudocritical temperature, °R

Tr= Reduced temperature, ⁰K

𝑇𝑟𝑖= Reduced temperature of component i, dimensionless

Vc = Critical molar volume, m3/kmol

𝑉𝑐𝑖= Critical molar volume of component i, ft3/lb-mol

𝑉𝑝𝑐= Pseudocritical molar volume, ft3/lb-mol

𝑉𝑠= Volume of solution, ft3

VI = Volume of unfiltered part, m3

𝑉𝐼,𝑖𝑛𝑖𝑡𝑖𝑎𝑙=Initial volume of unfiltered part, m3

VI = Volume of filtered part, m3

𝑉𝐼𝐼,𝐿= Volume of liquid phase in the filtered part, m3

𝑉𝐼𝐼,𝑉= Volume of vapor phase in the filtered part, m3

xi= Molar fraction of component i in the liquid phase, fraction

72

yi= Molar fraction of component i in the vapor phase, fraction

z= Compressibility factor

𝑧𝑖= Molar composition of component i in the mixture, fraction

𝑧𝐿= Compressibility factor of the liquid phase

𝑧𝑉= Compressibility factor of the vapor phase

Zi= Overall molar fraction of component i in the mixture, fraction

Zx = Molar fraction of unrestricted component x, fraction

Zy = Molar fraction of restricted component y, fraction

Greek letters

ij = Binary interaction parameter

σ= Filtration efficiency

𝜎𝑖 = Filtration efficiency of individual solute

𝜎𝑖∗= Filtration efficiency of individual solute, when no other solutes are present

𝜎𝑙= Filtration efficiency of lumped solutes

𝜋 = Osmotic pressure, psi

𝜋𝑟𝑒𝑎𝑙= Observed or realistic osmotic pressure, psi

𝜋𝑖𝑑𝑒𝑎𝑙=Theoretical osmotic pressure, psi

∆𝑛𝑟𝑖= Molar number change of restricted component i between two parts, mol

∆𝑛𝑚𝑖= Molar number change of unrestricted component i between two parts, mol

𝜙𝑖𝐿= Fugacity coefficient of component i in the liquid phase

𝜙𝑖𝑉= Fugacity coefficient of component i in the vapor phase

𝜌= Liquid density, lbm/ft3

73

𝜌𝐿= Liquid phase density, mole/cm3

𝜌𝑉= Vapor phase density, mole/cm3

𝜌𝑃𝑟= Reduced liquid density, dimensionless

= Acentric factor

𝛺𝑎= Peng-Robinson equation constant

𝛺𝑏= Peng-Robinson equation constant

𝜇= Fluid viscosity, cp

𝜇∗= Mixture viscosity at atmosphere pressure, cp

𝜇𝑖∗= Viscosity of component i at low pressure, cp

𝜉𝑖= Viscosity parameter of component i, cp-1

𝜉𝑚= Mixture viscosity parameter, cp-1

74

REFERENCES CITED

Brenneman, M.C., Smith, P. V. 1958. The Chemical Relationship between Crude Oils and Their Source Rocks. Habitat of Oil. American Association of Petroleum Geologist, Tulsa, Oklahoma, pp. 818-849.

Cath, T.Y., Childress, A.E, Elimelech, M. 2006. Forward Osmosis: Principles,

Applications, and Recent Developments. Journal of Membrane Science 281 (2006): 70-87. http://dx.doi.org/10.1016/j.memsci.2006.05.048.

Danesh, A. 1998. PVT and Phase Behavior of Petroleum Reservoir Fluids. Fisrt edition.

Chap. 8, 281-300. Elsevier Sience & Technology Books. Firoozabadi, A., Katz, D.L., Soroosh, H., and Sajjadian, V.A. 1988. Surface Tension of

Reservoir Crude Oil/Gas Systems Recognizing the Asphalt in the Heavy Fraction. SPE Reservoir Engineering 3 (1): 265-272. http://dx.doi.org/10.2118/13826-PA.

Garavito, A.M., Kooi, H., Neuzil, C.E. 2006. Numerical Modeling of Long-term in situ

Chemical Osmosis Experiment in the Pierre Shale, South Dakota. Advances in Water Resources 29, 481-492.

Geren, F., Firincioglu, T., Karacaer, C., Ozkan, E., and Ozgen, C. 2014. Modeling Flow

in Nanoporous, Membrane Reservoirs and Interpretation of Coupled Fluxes. Presented at the SPE Annual Technical Conference and Exhibition, Amsterdam, Netherlands, 27-29 October. SPE-170976-MS. http://dx.doi.org/10.2118/170976-MS

Geren, F. 2014. Modeling Flow in Nanoporous, Membrane Reservoirs and Interpretation

of Coupled Fluxes. MS Thesis, Colorado School of Mines, Golden, Colorado. Hunt, J.M., 1961. Distribution of Hydrocarbons in Sedimentary Rocks. Geochimica et

Cosmochimica Acta 22 (1), pp. 37-49 Hunt, J.M., Jameson, G. W. 1956. Oil and Organic Matter in Source Rocks of Petroleum.

American Association of Petroleum Geologist 40 (3), pp. 477-488. Keijzer, T. J. S. 2000. Chemical Osmosis in Natural Clayey Materials. Ph.D. thesis,

Utrecht University, Netherlands. Kuila, U. and Prasad, M. 2011. Surface Area and Pore-size Distribution in Clays and

Shales. Presented at the SPE Annual Technical Conference and Exhibition, Denver, Colorado, USA, 30 October - 2 November. SPE-146869-MS. http://dx.doi.org/10.2118/146869-MS

http://dx.doi.org/10.1016/j.memsci.2006.05.048

http://dx.doi.org/10.2118/13826-PA

http://dx.doi.org/10.2118/170976-MS



75

Lohrenz, J., Bray, B. G., Clark, C. R. 1964. Calculating Viscosities of Reservoir Fluids from Their Compositions, Journal of Petroleum Technology 16 (10): 1171-1176. SPE-915-PA. http://dx.doi.org/10.2118/915-PA

Magara, K., 1974. Compaction, Ion Filtration, and Osmosis in Shale and Their

Significance in Primary Migration. Bulletin of the American Association of Petroleum Geologist 58, pp. 283-290.

Marine, I.W. and Fritz, S.J. 1981. Osmotic Model to Explain Anomalous Hydraulic Heads,

Water Resources Research, 17 (1): 73-82. http://dx.doi.org/10.1029/WR017i001p00073

McKelvey, J. G. and Milne, J. H. 1960. The Flow of Salt Solutions through Compacted

Clay. Clays and Clay Minerals 9 (1): 248-259. http://dx.doi.org/10.1346/CCMN.1960.0090114

Mitchell, J. K. and Soga, K. 2005. Fundamentals of Soil Behavior, 3rd edition. Hoboken,

N.J.: John Wiley & Sons. Nelson, P. H. 2009. Pore-Throat Sizes in Sandstones, Tight Sandstones, and Shales.

Bulletin of the American Association of Petroleum Geologist 93 (3): 329-340. http://dx.doi.org/10.1306/10240808059

Neuzil, C. E. 2000. Osmotic Generation of ‘Anomalous’ Fluid Pressures in Geological

Environments Nature 403: 182-184. http://dx.doi.org/10.1038/35003174 Peng, D.Y. and Robinson, D.B. 1976. A New Two-Constant equation of State. Industrial

& Engineering Chemistry Fundamentals 15 (1): 59-64. http://dx.doi.org/10.1021/i160057a011

Prasad. M. 2012. Shales and imposters: understanding shales, organics, and self-

resourcing rocks. Presented at Southwest Louisiana Geophysical Society Luncheon. Lafayette, Louisiana, USA, October.

Revil, A. and Pessel, M. 2002. Electroosmotic Flow and the Validity of the Classical Darcy

Equation in Silty Shales. Geophysical Research Letters 29 (9), 1300, 10.1029/2001GL013480, pp. 14-1 – 14-4.

Sing, K. S. W. 1982. Physical and Biophysical Chemistry Division Commission on Colloid

and Surface Chemistry including Catalysis. Pure and Applied Chemistry 57 (4): 603-619. http://dx.doi.org/10.1351/pac198557040603

Staverman, A.J. 1952. Non-Equilibrium Thermodynamics of Membrane Processes.

Transactions of the Faraday Society. 48: 176-185. http://dx.doi.org/10.10 39/TF9524800176

http://dx.doi.org/10.2118/915-PA

http://dx.doi.org/10.1029/WR017i001p00073

http://dx.doi.prg/10.1346/CCMN.1960.0090114

http://dx.doi.org/10.1306/10240808059

http://dx.doi.org/10.1038/35003174

http://dx.doi.org/10.1021/i160057a011

http://dx.doi.org/10.1351/pac198557040603

http://dx.doi.org/10.10%2039/TF9524800176

http://dx.doi.org/10.10%2039/TF9524800176

76

van ’t Hoff, J.H. 1995. The Role of Osmotic Pressure in the Analogy between Solutions and Gasses. Journal of Membrane Science 100 (1): 39-44. http://dx.doi.org/10.1016/0376-7388(94)00232-N

Wang, L., Parsa, E., Gao, Y. et al. 2014. Experimental Study and Modeling of the Effect

of Nanoconfinement on Hydrocarbon Phase Behavior in Unconventional Reservoirs. Presented at the SPE Western North American and Rocky Mountain Joint Regional Meeting, Denver, Colorado, USA, 16-18 April. SPE-169581-MS. http://dx.doi.org/10.2118/169581-MS

Weaver, C.E. 1989. Continental Transport and Deposition. In Clays, Muds, and Shales,

first edition, Chapter IV, 189-270. Amsterdam, Netherlands: Elsevier Science Publishing Company Inc.

Weinaug, D.F. and Katz, D.L. 1943. Surface tensions of methane-propane mixtures.

Industrial and Engineering Chemistry 35 (2): 239-245. http://dx.doi.org/10.1021/ie50398a028

Wilson, G.M. 1968. A Modified Redlich-Kwong Equation-of-State, Application to General

Physical Data Calculations. Paper 15c presented at the AIChE Natl. Meeting, Cleveland, Ohio, 4-7 May.

Young, A. and Low, P. F. 1965. Osmosis in Argillaceous Rocks. Bulletin of the American

Association of Petroleum Geologist 49 (7): 1004-1008. Zhu, Z., Yin, X., Ozkan, E. 2015. Theoretical Investigation of the Effect of Membrane

Properties of Nanoporous Reservoirs on the Phase Behavior of Confined Light Oil. Presented at the SPE Annual Technical Conference and Exhibition, Houston, Texas, USA, 28-30 September. SPE-175152-MS. http://dx.doi.org/10.2118/175152-MS.

http://dx.doi.org/10.1016/0376-7388(94)00232-N


http://dx.doi.org/10.1021/ie50398a028


77

APPENDIX A

SIMULATION OF A PRESSURE DEPLETION PROCESS USING THE FUGACITY-

BASED FILTRATION EFFICIENCY

In Appendix A, the computational procedures and results for the simulation of a

pressure depletion process using the fugacity-based filtration efficiency are presented.

A.1 Solution of the Osmotic Pressure – Membrane Efficiency Equation

Figure A.1 Dual-pore system used to calculate membrane efficiency. (Geren 2014)

As shown in Figure A.1, we used a dual-pore system saturated with a binary

hydrocarbon mixture with components 1 and 2 to illustrate the calculation procedure for

the osmotic pressure – membrane efficiency equation. A nano-sized pore throat that acts

as a semi-permeable membrane connects the two pore systems. System I represents the

unfiltered part, and System II is the filtered part, in which the fluid can flow to production

wells without compositional change. The temperature T of the entire system is held

constant throughout the process. The pressure difference between two pores is the

osmotic pressure, ∆P = PI – PII. The entire system is saturated with a binary mixture that

contains a lighter hydrocarbon group 1 that can pass through the pore throat freely, and

78

a heavier group 2 that is restricted from freely moving between the two pore systems.

Both systems are kept in liquid state at their respective pressures PI and PII. In the

calculation procedure illustrated below, we use the osmotic pressure as a given and solve

for the membrane efficiency. This process can also be used reversely to compute the

osmotic pressure for a particular membrane efficiency.

Calculation Procedure: In System II, for a given composition ZII, we first perform an

EOS calculation at PII and T. When System II reaches equilibrium, we calculate the

fugacities of components 1 and 2, 𝑓𝐼𝐼1

𝐿 and 𝑓𝐼𝐼2

𝐿 , respectively. Then, an EOS calculation is

performed in System I at PI and T. We try to reach the equilibrium state between the two

pore systems by varying the composition of System I, ZI. At equilibrium, fugacity of the

light component 1 should be identical in both pore systems. However, due to the

membrane effect, the fugacity of the heavy component 2 will not be identical in the two

pore systems.

𝑓𝐼1

𝐿 = 𝑓𝐼𝐼1

𝐿 , 𝑓𝐼2

𝐿 ≠ 𝑓𝐼𝐼2

𝐿 (A.1)

After obtaining the fugacities of components 1 and 2 in each pore, calculate the

membrane efficiency from Eq. (A.2).

𝜔𝑓𝑖= 1 − (

𝑓𝐼𝐼,𝑖𝐿

𝑓𝐼,𝑖𝐿 ) (A.2)

The computational procedure described above is shown schematically in Figure

A.2. The computational algorithm used in this work has been developed based on a

modified Peng-Robinson EOS. Note that, the number of moles of components 1 and 2 in

each pore can also be calculated when the entire system reaches equilibrium. For

79

instance, the number of moles of component 1 in System I can be calculated by using Eq.

(A.3).

𝑛𝐼1= 𝑛𝐼 × 𝑍𝐼1

=𝑉𝐼𝑍𝐼1

𝑣𝐼,𝑚𝑖𝑥 (A.3)

Figure A.2 Computational procedure of solving the osmotic pressure-membrane

efficiency equation. (Geren et al. 2014)

where 𝑛𝐼 is the number of moles of hydrocarbons in System I, 𝑉𝐼 is the pore volume of

System I, 𝑍𝐼1 is the molar fraction of component 1 in System I and 𝑣𝐼_𝑚𝑖𝑥 is the calculated

molar volume of mixture in System I at corresponding temperature and pressure.

80

A.2 Coupling Membrane Filtration with Pressure Depletion

We modeled the constant-composition expansion of the entire system by

expanding the volume of System II and reducing its pressure. Based on the previous

osmotic pressure-membrane efficiency calculations, the number of moles of the

components and the compositions in both Systems I and II are known. The temperature

of the entire system is held constant at T. Because of pressure depletion, at the known

pressure 𝑃𝐼𝐼′ , which is below the bubble point, System II should consist of a liquid and

vapor mixture as illustrated in Figure A.3.

Figure A.3 Three-Pore system used to simulate the coupling of membrane filtration with

pressure depletion.

By performing a flash calculation and then an EOS calculation in System II at T, 𝑃𝐼𝐼′

and 𝑍𝐼𝐼′ , fugacities of components 1 and 2 in both liquid and vapor phases, 𝑓𝐼𝐼1

𝐿 , 𝑓𝐼𝐼2

𝐿 , 𝑓𝐼𝐼1

𝑉 ,

and 𝑓𝐼𝐼2

𝑉 , can now be calculated. Then, an EOS calculation is performed in System I at T

and assumed 𝑃𝐼′ and 𝑍𝐼

′ . The pressure 𝑃𝐼′ and composition 𝑍𝐼

′ are varied to reach the

equilibrium state between the two pores by satisfying the following equations.

𝑓𝐼1



𝑉 (A.4)

81

𝑓𝐼𝐼2

𝐿 =𝑓𝐼𝐼2

𝐿

1−𝜔𝑓 (A.5)

Also, because we are modeling a closed system, the change of the number of moles of

unrestricted component 1 in Systems I and II should be zero when the entire system

reaches an equilibrium state.

∆𝑛𝐼1= ∆𝑛𝐼𝐼1

= 0 (A.6)

∆𝑛1 is the change of number of moles of unrestricted component 1 in System I between

two successive calculations, and ∆𝑛𝐼𝐼1 is the molar number change of component 1 in

System II. If ∆𝑛𝐼1 and ∆𝑛𝐼𝐼1

are nonzero, then the compositions of both Systems I and II

are updated, and the procedure is repeated until Eqs. (A.4), (A.5), and (A.6) are satisfied

simultaneously. For instance, the composition of System I is updated by using Eq. (A.7)

and Eq. (A.8).

𝑍𝐼1

′ =𝑛𝐼1+∆𝑛𝐼1

𝑛𝐼1+∆𝑛𝐼1+𝑛𝐼2

(A.7)

𝑍𝐼2

′ =𝑛𝐼2

𝑛𝐼1+∆𝑛𝐼𝐼1+𝑛𝐼2

(A.8)

𝑛𝐼1 is the number of moles of unrestricted component 1 in System I from previous

calculation, ∆𝑛𝐼1 is the molar number change between the current and previous

calculations, and 𝑛𝐼2 is the number of moles of restricted component 2 in System I, which

stays constant. The computational procedure of coupling of membrane filtration with

pressure depletion is shown in Figure A.4.

82

Figure A.4 Computational procedure of coupling membrane filtration with pressure depletion.

A.3 Simulation Results

In this section, we present the results of the simulation of a pressure depletion of a

light-oil confined in the pore systems with and without membrane properties at different

depletion pressures using the fugacity-based filtration efficiency. The light oil consists of

nC4, nC10, nC16 and C24. Among all the components, C24 is the only one restricted from

moving through the pore throats, therefore, is subjected to membrane filtration. The other

components can move freely between the pores without hindrance. Table A.1 shows the

83

parameters of the components, including critical properties, acentric factors, molecular

weights and binary interaction parameters.

The temperature of the pore systems is held constant at 360.9 ⁰K. The initial

pressures of Systems I and II are set as 5000 psi and 4000 psi, respectively (the osmotic

pressure between the two systems is 1000 psi). The initial volumes of both systems are

set as 5 × 10−25 m3. System II is initially filled with 30% nC4, 30% nC10, 20% nC16, and

20% C24 (mole fraction). By performing the osmotic pressure – membrane efficiency

calculations, we can obtain the composition of System I that corresponds to a membrane

efficiency of 𝜔𝑓 = 0.8476. The viscosities of Systems I and II are calculated by using the

Lohrenz correlation (Lohrenz et al. 1964). The simulation parameters and results are

presented in Table A.2.

Table A.1 Thermodynamic model parameters of the components in the light oil


Tc (⁰K) 425.2 617.6 717 823.2

Pc (psi) 551.1 305.7 205.7 181.9

Vc (m3/kmol) 0.255 0.603 0.956 1.17

ω 0.193 0.49 0.742 0.940079

MW (g/mol) 58.124 142.286 226.448 324

δij

nC4 0 0.012228 0.028461 0.037552

nC10 0.012228 0 0.00353 0.007281

nC16 0.028461 0.00353 0 0.00068

C24 0.037552 0.007281 0.00068 0

The results shown in Table A.2 serve as the initial condition for the constant-

composition expansion. Table A.3 presents the resulting fluid properties when the

pressure of System II is reduced to 45 psi. By comparing the number of moles of

components in the two pore systems before and after pressure depletion, it is easy to see

that the number of moles of C24 in both pore systems is kept constant, indicating that C24

84

is restricted from moving between the pores. Because of the pressure reduction, some

light components in System II vaporize into the gas phase, leading to a compositional

change of the liquid phase in System II. As a result, a new osmotic pressure (997 psi) is

established. In this process, the light components, which are unrestricted, move from

System I (unfiltered part) to System II (filtered part), to re-establish the thermodynamic

equilibrium between the liquid phases of the two systems.

Table A.2 Simulation parameters and results before pressure depletion






22.52 17.65 8.99 50.82 30 30 20 20


3.55 2.78 1.42 8.01 5.92 5.92 3.95 3.95




Density (kg/m3) 703.3 672.0

Membrane Efficiency 𝜔𝑓 0.8476

85

Table A.3 Fluid properties when the pressure of System II is reduced to 45 psi. Membrane effect is implemented between System I and System II

Properties System I (Unfiltered) System II (Filtered)




2.73 2.76 1.40 8.01 6.74 5.94 3.96 3.95


18.34 18.53 9.40 53.75 32.72 28.86 19.24 19.17




Vapor phase composition (mol%) -- 99.21 0.78 6.5E-3 3.7E-05








86

To illustrate the differences caused by the membrane, we simulate a constant

composition expansion process without the membrane. The composition of the liquid

comes from a combination of the liquids in Systems I and II (c.f. Table A.2). The properties

of the fluids at 45 psi are listed in Table A.4. As we compare Table A.3 and A.4, it is

noticed that both the system composition and the liquid phase composition without

membrane filtration lie between the values of the corresponding compositions of Systems

I and II with membrane filtration. This result illustrates the effect of the internal membrane

barrier. Compared with the system without the membrane, at 45 psi, the liquid that

remains in System II has a lower C24 composition. This result verifies our expectation that

the internal filtration may reduce the molecular weight of the produced hydrocarbons.

Table A.5 through A.8 present the simulation results for pore systems with and

without membrane filtration when the pressure is further reduced to 35 psi and 25 psi.

As the pressure of System II decreases, the mole fraction of the liquid phase in

System II decreases, which indicates that more of the light components have vaporized

into the gas phase. The vaporization of the light components increases the imbalance

between the liquid phases in System I and System II. Accordingly, the osmotic pressure

between the two systems increases first to 1016 psi and then 1035 psi to maintain the

equilibrium (Figure A.5). Also, with the vaporization of the light components, the liquid

phases on both sides of the membrane become heavier, leading to increased densities

and molar volumes. It is surprising that, as the pressure of System II decreases, the

pressure of System I has to increase to maintain the equilibrium due to increased osmotic

pressure. This result, which seems to be counterintuitive, is due to the vaporization of the

light components in System II.

87

Table A.4 Fluid properties from a constant composition expansion at 45 psi without membrane filtration


Pressure (psi) 45


System composition (nC4-nC10-nC16-C24, mol%) 26.68 24.52 15.11 33.68


Liquid phase (nC4-nC10-nC16-C24, mol%) 24.39 25.28 15.59 34.75


Vapor phase (nC4-nC10-nC16-C24, mol%) 99.37 0.63 4.7E-03 5.7E-05


Liquid phase fraction (mol%) 96.9





Membrane Efficiency 𝜔𝑓 0

88

Table A.5 Fluid Properties when the pressure of System II is reduced to 35 psi. Membrane effect is included





2.03 2.79 1.41 8.01 7.44 5.91 3.95 3.95


14.27 19.60 9.93 56.20 35.01 27.82 18.58 18.58












89

Table A.6 Fluid properties when the pressure of System II is reduced to 35 psi. Membrane effect is not included


Pressure (psi) 35


System composition (nC4-nC10-nC16-C24, mol%) 26.68 24.52 15.11 33.68












90

Table A.7 Fluid properties when the pressure of System II is reduced to 25 psi. Membrane effect is included





1.39 2.82 1.42 8.01 8.08 5.88 3.94 3.95


10.17 20.70 10.43 58.70 36.99 26.91 18.03 18.07












91

Table A.8 Fluid properties when the pressure of System II is reduced to 25 psi. Membrane effect is not included


Pressure (psi) 25


System composition (nC4-nC10-nC16-C24, mol %) 26.68 24.52 15.11 33.68












92

Figure A.5 Osmotic pressure at different depletion pressures.

970

980

990

1000

1010

1020

1030

1040

20 25 30 35 40 45 50 55

Osm

otic p

ressu

re, p

si

Final pressure of System 2, psi

93

APPENDIX B

SIMULATION CASE WITH A FILTRATION EFFICIENCY OF [0.75, 0.9]

In Appendix B, we present the simulation results of a pressure depletion process

starting from an initial-equilibrium stage for a porous medium with a filtration efficiency of

[0.75, 0.9].

Table B.1 Thermodynamic parameters of components in the light oil

Table B.2 Initial state parameters






10.0 10.0 30.0 50.0 30.0 30.0 20.0 20.0


Tc (⁰K) 425.2 617.6 717 823.2

Pc (psi) 551.1 305.7 205.7 181.9

Vc (m3/kmol) 0.255 0.603 0.956 1.17

ω 0.193 0.49 0.742 0.940079

MW (g/mol) 58.124 142.286 226.448 324

δij

nC4 0 0.012228 0.028461 0.037552

nC10 0.012228 0 0.00353 0.007281

nC16 0.028461 0.00353 0 0.00068

C24 0.037552 0.007281 0.00068 0

94

Table B.3 The initial-equilibrium state before pressure depletion






15.75 10.64 27.91 45.70 26.52 30.57 21.12 21.79


2.352 1.588 4.166 6.823 4.997 5.761 3.980 4.107




Density (kg/m3) 703.4 674.1

95

Table B.4 Post-initial equilibrium state when the pressure of the filtered part is reduced to 30 psi. Ideal membrane, σ = [1, 1]






1.275 1.508 4.166 6.823 6.074 5.841 3.980 4.107


9.26 10.95 30.25 49.54 30.36 29.20 19.90 20.53













96

Table B.5 Post-initial equilibrium state when the pressure of the filtered part is reduced to 30 psi. Non-ideal membrane, σ = [0.75, 0.9]






1.383 1.710 4.144 6.691 5.966 5.639 4.003 4.240


9.93 12.28 29.75 48.04 30.06 28.41 20.17 21.36













97

Table B.6 Post-initial equilibrium state when the pressure of the filtered part is reduced to 30 psi. Non-selective membrane, σ = [0, 0]


Pressure (psi) 30

Temperature (°K) 360.9

Composition (nC4-nC10-nC16-C24, mol%) 21.76 21.76 24.12 32.36


Liquid phase composition (mol%) 16.33 23.23 25.81 34.63

Liquid phase apparent molecular weight 213.2

Vapor phase composition (mol%) 99.18 0.81 0.01 7.118E-05

Vapor phase apparent molecular weight 58.8

Liquid phase fraction (mol%) 93





98

Table B.7 Overall and individual filtration efficiencies for every case

Case I Case II Case III

Individual Filtration Efficiency σ nC4 nC10 nC16 C24 nC4 nC10 nC16 C24 nC4 nC10 nC16 C24

0.0 0.0 1.0 1.0 0.0 0.0 0.7508 0.9074 0.0 0.0 0.0 0.0

Overall Filtration Efficiency,

from 𝜎 = (𝑃𝐹,𝑟𝑒𝑎𝑙

𝑃𝐹,𝑖𝑑𝑒𝑎𝑙 )𝐽𝑣=0

1.0 0.8711 0.0

Overall Filtration Efficiency, from Eq. (4.9) 1.0 0.8998 0.0

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